Go Math solutions for Class 6 Maths Provide detailed explanations for all the questions provided in the HMH Go Math. We provide topic wise Go Math Grade 6 Answer Key Chapter 11 Surface Area and Volume to help the students clear their doubts by offering an understanding of concepts in depth. You can practice different types of questions in Go Math Grade 6 Answer Key Chapter 11 Surface Area and Volume.

## Go Math Grade 6 Answer Key Chapter 11 Surface Area and Volume

Download HMH Go Math Grade 6 Answer Key Chapter 11 Surface Area and Volume and learn offline. With the help of these Go Math 6th Grade Solution Key Chapter 11 Surface Area and Volume, you can score good marks in the exams. The topics include 3-D figures and Nets, Explore Surface Area Using Nets, Surface Area of Prisms, and so on. This will also help to build a strong foundation of all these concepts for secondary level classes.

**Lesson 1: Three-Dimensional Figures and Nets**

- Share and Show – Page No. 599
- Problem Solving + Applications – Page No. 600
- Three-Dimensional Figures and Nets – Page No. 601
- Lesson Check – Page No. 602

**Lesson 2: Investigate • Explore Surface Area Using Nets**

- Share and Show – Page No. 605
- What’s the Error? – Page No. 606
- Explore Surface Area Using Nets – Page No. 607
- Lesson Check – Page No. 608

**Lesson 3: Algebra • Surface Area of Prisms**

- Share and Show – Page No. 611
- Unlock the Problem – Page No. 612
- Surface Area of Prisms – Page No. 613
- Lesson Check – Page No. 614

**Lesson 4: Algebra • Surface Area of Pyramids**

- Share and Show – Page No. 617
- Problem Solving + Applications – Page No. 618
- Surface Area of Pyramids – Page No. 619
- Lesson Check – Page No. 620

** Mid-Chapter Checkpoint**

**Lesson 5: Investigate • Fractions and Volume**

- Share and Show – Page No. 625
- Problem Solving + Applications – Page No. 626
- Fractions and Volume – Page No. 627
- Lesson Check – Page No. 628

**Lesson 6: Algebra • Volume of Rectangular Prisms**

- Share and Show – Page No. 631
- Aquariums – Page No. 632
- Volume of Rectangular Prisms – Page No. 633
- Lesson Check – Page No. 634

**Lesson 7: Problem Solving • Geometric Measurements**

- Share and Show – Page No. 637
- On Your Own – Page No. 638
- Problem Solving Geometric Measurements – Page No. 639
- Lesson Check – Page No. 640

**Chapter 11 Review/Test**

- Chapter 11 Review/Test – Page No. 641
- Page No. 642
- Page No. 643
- Page No. 644
- Page No. 645
- Page No. 646

### Share and Show – Page No. 599

**Identify and draw a net for the solid figure.**

Question 1.

Answer: The base Square or Rectangle, and lateral faces are Triangle and the figure is a Square pyramid or Rectangular pyramid.

Explanation:

Question 2.

Answer: Cube or Rectangular prism.

Explanation: The base is a square or rectangle and lateral faces are squares are rectangle. The figure is a Cube or Rectangular prism.

**Identify and sketch the solid figure that could be formed by the net.**

Question 3.

Answer: Triangular pyramid.

Explanation: The net has four triangles, so it is a triangular pyramid.

Question 4.

Answer: Cube

Explanation: The net has six squares.

**On Your Own**

**Identify and draw a net for the solid figure.**

Question 5.

Answer: Triangular prism.

Explanation: The base is a rectangle and the lateral faces are triangle and rectangles, so it is a triangular prism.

Question 6.

Answer: Rectangular Prism.

Explanation: The base is a rectangle and the lateral faces are squares and rectangles. And it is a Rectangular prism.

### Problem Solving + Applications – Page No. 600

**Solve.**

Question 7.

The lateral faces and bases of crystals of the mineral galena are congruent squares. Identify the shape of a galena crystal.

Answer: Cube

Explanation: The shape of the galena is Cube.

Question 8.

Rhianon draws the net below and labels each square. Can Rhianon fold her net into a cube that has letters A through G on its faces? Explain.

Answer: No, she cannot fold her net into a cube. Rhianon’s net has seven squares but there are only six squares in a net of a cube.

Question 9.

Describe A diamond crystal is shown. Describe the figure in terms of the solid figures you have seen in this lesson.

Answer: We can see that Diamond crystal consists of two square pyramids with congruent bases and the pyramids are reversed and placed base to base.

Question 10.

Sasha makes a triangular prism from paper.

The bases are _____.

The lateral faces are _____.

Answer:

The bases are Triangle

The lateral faces are Rectangle

### Three-Dimensional Figures and Nets – Page No. 601

**Identify and draw a net for the solid figure.**

Question 1.

Answer: Rectangular Prism

Explanation:

Question 2.

Answer: Cube, Rectangular prism

Explanation:

Question 3.

Answer: Square Pyramid

Explanation:

Question 4.

Answer: Triangular Prism

Explanation:

**Problem Solving**

Question 5.

Hobie’s Candies are sold in triangular-pyramidshaped boxes. How many triangles are needed to make one box?

Answer: 4

Explanation: As triangled pyramids have four faces.

Question 6.

Nina used plastic rectangles to make 6 rectangular prisms. How many rectangles did she use?

Answer: 36

Explanation:

Question 7.

Describe how you could draw more than one net to represent the same three-dimensional figure. Give examples.

Answer:

Explanation:

### Lesson Check – Page No. 602

Question 1.

How many vertices does a square pyramid have?

Answer: 5

Explanation:

Question 2.

Each box of Fred’s Fudge is constructed from 2 triangles and 3 rectangles. What is the shape of each box?

Answer: Triangular Prism

Explanation:

**Spiral Review**

Question 4.

A hot-air balloon is at an altitude of 240 feet. The balloon descends 30 feet per minute. What equation gives the altitude y, in feet, of the hot-air balloon after x minutes?

Answer: Y= 240- 30X.

Explanation: Given altitude Y, and the ballon was descended 30 feet per minute. So the equation is Y= 240- 30X.

Question 6.

Alexis draws quadrilateral STUV with vertices S(1, 3), T(2, 2), U(2, –3), and V(1, –2). What name best classifies the quadrilateral?

Answer: Parallelogram

Explanation:

### Share and Show – Page No. 605

**Use the net to find the surface area of the prism.**

Question 1.

Answer:

Explanation: First we must find the area of each face

A= 4×3= 12

B= 4×3= 12

C= 5×4= 20

D= 5×4= 20

E= 5×3= 15

F= 5×3= 15

So, the surface area is 12+12+20+20+15+15= 94 cm^{2}

**Find the surface area of the rectangular prism.**

Question 2.

Answer: 222 cm^{2}

Explanation: Area of a rectangular prism is 2(wl+hl+hw) = 2(7×9+ 3×9+ 3×7)

= 2(63+27+21)

= 2(111)

= 222 cm^{2}

Question 3.

Answer:

Explanation: Area of a rectangular prism is 2(wl+hl+hw) = 2(10×10+ 10×10+ 10×10)

= 2(100+100+100)

= 2(300)

= 600 cm^{2}

Question 4.

Answer: 350 cm^{2}

Explanation: Area of a rectangular prism is 2(wl+hl+hw) = 2(15×5+ 5×5+ 15×5)

= 2(75+25+75)

= 2(175)

= 350 cm^{2}

**Problem Solving + Applications**

Question 6.

Darren is painting a wooden block as part of his art project. The block is a rectangular prism that is 12 cm long by 9 cm wide by 5 cm high. Describe the rectangles that make up the net for the prism.

Answer:

Question 7.

In Exercise 6, what is the surface area, in square meters, that Darren has to paint?

Answer: 416 cm^{2}

Explanation: Surface area = 2(wl+hl+hw)

= 2(9×12+5×12+ 5×9)

= 2(108+60+45)

= 2(213)

= 416 cm^{2}

### What’s the Error? – Page No. 606

Question 8.

Emilio is designing the packaging for a new MP3 player. The box for the MP3 player is 5 cm by 3 cm by 2 cm. Emilio needs to find the surface area of the box.

Look at how Emilio solved the problem. Find his error.

STEP 1: Draw a net.

STEP 2: Find the areas of all the faces and add them.

Face A: 3 × 2 = 6 cm^{2}.

Face B: 3 × 5 = 15 cm^{2}.

Face C: 3 × 2 = 6 cm^{2}.

Face D: 3 × 5 = 15 cm^{2}.

Face E: 3 × 5 = 15 cm^{2}.

Face F: 3 × 5 = 15 cm^{2}.

The surface area is 6 + 15 + 6 + 15 + 15 + 15 = 72 cm^{2}.

Correct the error. Find the surface area of the prism.

Answer: Emilio drew the net incorrectly Face D and F should have been 2 cm by 5 cm, not 3 cm by 5 cm

Explanation:

Face A: 3×2= 6 cm^{2}

Face B: 3×5= 15 cm^{2}

Face C: 3×2= 6 cm^{2}

Face D: 2×5= 10 cm^{2}

Face E: 3×5= 15 cm^{2}

Face F: 2×5= 10 cm^{2}

So, the surface area of the prism area is 6+15+6+10+15+10= 62 cm^{2.}

Question 9.

For numbers 9a–9d, select True or False for each statement.

9a. The area of face A is 10 cm^{2}.

9b. The area of face B is 10 cm^{2}.

9c. The area of face C is 40 cm^{2}.

9d. The surface area of the prism is 66 cm^{2}.

9a. The area of face A is 10 cm^{2}.

Answer: True

Explanation: The area of face A is 2×5= 10 cm^{2}.

9b. The area of face B is 10 cm^{2}.

Answer: False

Explanation: The area of face B is 2×8= 16 cm^{2}.

9c. The area of face C is 40 cm^{2}.

Answer: The area of face C is 8×5= 40 cm^{2}.

9d. The surface area of the prism is 66 cm^{2}.

Answer: 160 cm^{2}.

Explanation: The surface area of the prism is

= 2×10+2×10+2×40

= 20+20+80

= 160 cm^{2}.

### Explore Surface Area Using Nets – Page No. 607

**Use the net to find the surface area of the rectangular prism.**

Question 1.

_______ square units

Answer: 52 square units.

Explanation:

The area of face A is 6 squares.

The area of face B is 8 squares.

The area of face C is 6 squares.

The area of face D is 12 squares.

The area of face E is 8 squares.

The area of face F is 12 squares.

The surface area is 6+8+6+12+8+12= 52 square units.

Question 2.

_______ square units

Answer: 112 square units.

Explanation:

The area of face A is 16 squares.

The area of face B is 8 squares.

The area of face C is 32 squares.

The area of face D is 16 squares.

The area of face E is 32 squares.

The area of face F is 8 squares.

The surface area is 112 square units.

Question 3.

Answer: 102 mm^{2}

Explanation: Area= 2(wl+hl+hw)

= 2(3×7+3×7+3×3)

= 2(21+21+9)

= 2(51)

= 102 mm^{2}

Question 4.

_______ in.^{2}

Answer: 58 in.^{2}

Explanation: Area= 2(wl+hl+hw)

= 2(5×1+ 4×1+ 4×5)

= 2(5+4+20)

= 2(29)

= 58 in.^{2}

Question 5.

_______ ft^{2}

Answer: 77 ft^{2}

Explanation: Area= 2(wl+hl+hw)

= 2(6.5×2+3×2+3×6.5)

= 2(13+6+19.5)

= 2(38.5)

= 77 ft^{2}

**Problem Solving**

Question 6.

Jeremiah is covering a cereal box with fabric for a school project. If the box is 6 inches long by 2 inches wide by 14 inches high, how much surface area does Jeremiah have to cover?

_______ in.^{2}

Answer: 248 in.^{2}

Explanation: Surface area of a cereal box is 2(wl+hl+hw)

= 2(2×6+14×6+14×2)

= 2(12+84+28)

= 2(124)

= 248 in.^{2
}

Question 7.

Tia is making a case for her calculator. It is a rectangular prism that will be 3.5 inches long by 1 inch wide by 10 inches high. How much material (surface area) will she need to make the case?

_______ in.^{2}

Answer: 97 in.^{2}

Explanation: Surface Area= 2(wl+hl+hw)

= 2(1×3.5+ 10×3.5+ 10×1)

= 2(3.5+35+10)

= 2(48.5)

= 97 in.^{2}

Question 8.

Explain in your own words how to find the surface area of a rectangular prism.

Answer: To find the surface area we must know the width, length, and height of the prism and then we can apply the formula which is

Surface area= 2(width ×length)+ 2(length×height)+ 2(height×width)

= 2(width ×length+ length×height+ 2(height×width)

### Lesson Check – Page No. 608

**Spiral Review**

Question 4.

The table shows the number of mixed CDs y that Jason makes in x hours.

Which equation describes the pattern in the table?

Answer: y= 5x

Explanation:

y/x = 10/2= 15/4= 3

y= 5x

The pattern is y is x multiplied by 5.

Question 6.

Boxes of Clancy’s Energy Bars are rectangular prisms. How many lateral faces does each box have?

Answer: 4

Explanation: Lateral faces are not included in the bases, so the rectangular prism has 4.

### Share and Show – Page No. 611

**Use a net to find the surface area.**

Question 1.

_______ ft^{2}

Answer: 24 ft^{2}

Explanation: The area of each face is 2 ft×2 ft= 4 ft and the number of faces is 6, so surface area is 6×4= 24 ft^{2}

Question 2.

Answer: 432 cm^{2}

Explanation:

The area of face A is 16×6= 96 cm^{2
}The area of face B is 16×8= 128 cm^{2}

The area of face C and D is 1/2 × 6×8= 24 cm^{2}

The area of face E is 16×10= 160 cm^{2}

The surface 96+128+2×24+160= 432 cm^{2}

Question 3.

_______ in.^{2}

Answer: 155.5 in.^{2}

Explanation:

The area of face A and E is 8 ½ × 3½

= 17/2 × 7/2

= 119/4

= 29.75 in.^{2}

The area of face B and F is 8 ½×4

= 17 ½ × 4

= 34 in.^{2
}The area of face C and D is 3 ½×4

7/2 × 4= 14 in.^{2
}The surface area is 2×29.75+2×34+2×14

= 59.5+68+28

= 155.5 in.^{2}

**On Your Own**

**Use a net to find the surface area.**

Question 4.

_______ m^{2}

Answer:

Explanation:

The area of face A and E is 8×3= 24 m^{2
}The area of face B and F is 8×5= 40 m^{2
}The area of face C and D is 3×5= 15 m^{2
}The surface area is 2×24+2×40+2×15

= 48+80+30

= 158 m^{2}

Question 5.

_______ \(\frac{□}{□}\) in.^{2}

Answer:

Explanation:

The area of each face is 7 1/2 × 7 1/2

= 15/2 × 15/2

= 225/4 in.^{2}

The no.of faces are 6 and the surface area is 6× 225/4

= 675/4

= 337 1/2 in.^{2}

Question 6.

Attend to Precision Calculate the surface area of the cube in Exercise 5 using the formula S = 6s^{2}. Show your work.

Answer: 337 1/2 in.^{2}

Explanation: As S= s^{2}

= 6(7 1/2)^{2}

= 6(15/2)^{2}

= 6(225/4)

= 675/2

= 337 1/2 in.^{2}

### Unlock the Problem – Page No. 612

Question 7.

The Vehicle Assembly Building at Kennedy Space Center is a rectangular prism. It is 218 m long, 158 m wide, and 160 m tall. There are four 139 m tall doors in the building, averaging 29 m in width. What is the building’s outside surface area when the doors are open?

a. Draw each face of the building, not including the floor.

Answer:

Question 7.

b. What are the dimensions of the 4 walls?

Answer: The 2 walls measure 218 m ×160 m and 2 walls measure by 158 m×160 m.

Question 7.

c. What are the dimensions of the roof?

Answer: The dimensions of the roof are 218 m×158 m.

Question 7.

d. Find the building’s surface area (not including the floor) when the doors are closed.

_______ m^{2}

Answer: 1,54,764 m^{2}

Explanation:

The area of two walls is 218×160= 34,880 m^{2}

The area of the other two walls is 158×160= 25,280 m^{2
}The area of the roof 158×218= 34,444 m^{2
}The surface area is 2× 34,880+ 2× 25,280+ 34,444

= 69,760+ 50,560+ 34,444

= 1,54,764 m^{2}

Question 7.

e. Find the area of the four doors.

_______ m^{2}

Answer: 16,124 m^{2}

Explanation: The area of a door is 139×29 = 4031 m^{2}

And the area of 4 doors is 4×4031= 16,124 m^{2}

Question 7.

f. Find the building’s surface area (not including the floor) when the doors are open.

_______ m^{2}

Answer: 1,38,640 m^{2}

Explanation: The building’s surface area (not including the floor) when the doors are open is

1,54,764 – 16,124= 1,38,640 m^{2}

### Surface Area of Prisms – Page No. 613

**Use a net to find the surface area.**

Question 1.

_______ cm^{2}

Answer: 104 cm^{2}

Explanation: Surface area= 2(wl+hl+hw)

= 2(6×5+2×5+2×6)

= 2(30+10+12)

= 2(52)

= 104 cm^{2}

Question 2.

_______ in.^{2}

Answer: 118 in.^{2}

Explanation: Surface area= 2(wl+hl+hw)

= 2(3.5×4+6×4+6×3.5)

= 2(59)

= 118 in.^{2}

Question 3.

_______ ft^{2}

Answer: 486 ft^{2}

Explanation: Surface area= 2(wl+hl+hw)

= 2(9×9+9×9+9×9)

= 2(81+81+81)

= 2(243)

= 486 ft^{2}

Question 4.

_______ cm^{2}

Answer: 336 cm^{2.}

Explanation: Area = 1/2 bh

= 1/2 (6)(8)

= 3×8

= 24.

As there are 2 triangles, so 2×24= 48.

Surface Area= (wl+hl+hw)

= (6×12+8×12+12×10)

= 228

Total Surface area = 228+48

= 336 cm^{2}

**Problem Solving**

Question 5.

A shoe box measures 15 in. by 7 in. by 4 \(\frac{1}{2}\) in. What is the surface area of the box?

_______ in.^{2}

Answer: 408 in.^{2}

Explanation:

The area of two faces is 15×7= 105 in.^{2
}The area of two faces is 15× 4 1/2

= 15 × 9/2

= 15 × 4.5

= 67.5 in.^{2}

The area of two faces is 7× 4 1/2

= 7× 9/2

= 7× 4.5

= 31.5 in.^{2
}The surface area is 2×105+ 2×67.5+ 2×31.5

= 210+ 135+ 63

= 408 in.^{2}

Question 7.

Explain why a two-dimensional net is useful for finding the surface area of a three-dimensional figure.

Answer: Two-dimensional net is useful because by using a two-dimensional net you can calculate the surface area of each face and add them up to find the surface area of the three-dimensional figure.

### Lesson Check – Page No. 614

Question 1.

What is the surface area of a cubic box that contains a baseball that has a diameter of 3 inches?

_______ in.^{2}

Answer: 54 in.^{2}

Explanation:

The area of each face is 3×3= 9 in.^{2}

The number of faces for a cubic box is 6 in.^{2}

The surface area of box that contains a baseball is 6×9= 54 in.^{2}

**Spiral Review**

Question 3.

Detergent costs $4 per box. Kendra graphs the equation that gives the cost y of buying x boxes of detergent. What is the equation?

Answer: Y= 4X.

Explanation: The total price is Y and the price is equal to 4 × X, and X is the number of boxes that Kendra buys. So the equation is Y=4X.

Question 5.

City Park is a right triangle with a base of 40 yd and a height of 25 yd. On a map, the park has a base of 40 in. and a height of 25 in. What is the ratio of the area of the triangle on the map to the area of City Park?

Answer: 1296:1.

Explanation:

Area= 1/2 bh

= 1/2 (40)(25)

= (20)(25)

= 500 yd^{2}

So area of city park is 500 yd^{2}

Area= 1/2 bh

= 1/2 (40)(25)

= (20)(25)

= 500 in^{2
}So area on the map is 500 in

as 1 yd^{2}= 1296 in^{2}

So 500 in^{2} = 500×1296

= 648,000

So, the ratio of the area of the triangle on the map to the area of City Park is 648,000:500

= 1296:1.

Question 6.

What is the surface area of the prism shown by the net?

Answer: 72 square units.

Explanation:

The area of two faces is 18 squares

The area of two faces is 6 squares

The area of two faces is 12 squares

So the surface area is 2×18+ 2×6+ 2×12

= 72 square units.

### Share and Show – Page No. 617

Question 1.

Use a net to find the surface area of the square pyramid.

_______ cm^{2}

Answer: 105 cm^{2}

Explanation:

Area of the base 5×5= 25 ,

and area of one face is 1/2 × 5 × 8

= 5× 4

= 20 cm^{2}

The surface area of a pyramid is 25+ 4×20

= 25+80

= 105 cm^{2}

Question 2.

A triangular pyramid has a base with an area of 43 cm^{2} and lateral faces with bases of 10 cm and heights of 8.6 cm. What is the surface area of the pyramid?

_______ cm^{2}

Answer: 172 cm^{2}

Explanation:

The area of one face is 1/2×10×8.6

= 5×8.6

= 43 cm^{2}

The surface area of the pyramid is 43+3×43

= 43+ 129

= 172 cm^{2}

**On Your Own**

**Use a net to find the surface area of the square pyramid.**

Question 4.

_______ ft^{2}

Answer: 208 ft^{2}

Explanation:

The area of the base is 8×8= 64

The area of one face is 1/2 ×8×9

= 36 ft^{2}

The surface area of the pyramid is 64+4×36

= 64+144

= 208 ft^{2}

Question 5.

_______ cm^{2}

Answer: 220 cm^{2}

Explanation:

The area of base is 10×10= 100

The area of one place is 1/2×10×6

= 10×3

= 30

The surface area of the pyramid is 100+4×30

= 100+120

= 220 cm^{2}

Question 6.

_______ in.^{2}

Answer: 264 in.^{2}

Explanation:

The area of the base is 8×8= 64

The area of one face is 1/2×8×12.5

= 4×12.5

= 50 in.^{2}

The surface area of the pyramid is 64+ 4×50

= 64+200

= 264 in.^{2}

Question 7.

The Pyramid Arena is located in Memphis, Tennessee. It is in the shape of a square pyramid, and the lateral faces are made almost completely of glass. The base has a side length of about 600 ft and the lateral faces have a height of about 440 ft. What is the total area of the glass in the Pyramid Arena?

_______ ft^{2}

Answer: 5,28,000 ft^{2}

Explanation:

The area of one face is 1/2×600×440= 1,32,000 ft^{2}

The surface of tha lateral faces is 4× 1,32,000= 5,28,000 ft^{2}

So, the total area of the glass in the arena is 5,28,000 ft^{2}

### Problem Solving + Applications – Page No. 618

**Use the table for 8–9.**

Question 8.

The Great Pyramids are located near Cairo, Egypt. They are all square pyramids, and their dimensions are shown in the table. What is the lateral area of the Pyramid of Cheops?

_______ m^{2}

Answer: 82,800 m^{2}

Explanation:

The area of one face is 1/2×230×180

= 230×90

= 20,700 m^{2}

The lateral area of the pyramid of Cheops is 4×20,700= 82,800 m^{2}

Question 9.

What is the difference between the surface areas of the Pyramid of Khafre and the Pyramid of Menkaure?

_______ m^{2}

Answer: 93,338 m^{2}

Explanation:

The area of the base is 215×215= 46,225

The area of one face is 1/2×215×174

= 215× 87

18,705 m^{2}

The surface area of Pyramid Khafre is= 46,225+4×18,705

= 46,225+ 74820

= 121,045 m^{2}

The area of the base 103×103= 10,609

The area of one face is 1/2×103×83

= 8549÷2

= 4274.4 m^{2}

The surface area of the Pyramid of Menkaure is 10,609+4×4274.5

= 10,609+ 17,098

= 27,707 m^{2}

The difference between the surface areas of the Pyramid of Khafre and the Pyramid of Menkaure

= 121,405-27,707

= 93,338 m^{2}

Question 10.

Write an expression for the surface area of the square pyramid shown.

Answer: 6x+9 ft^{2.}

Explanation: The expression for the surface area of the square pyramid is 6x+9 ft^{2.}

Question 12.

Jose says the lateral area of the square pyramid is 260 in.^{2}. Do you agree or disagree with Jose? Use numbers and words to support your answer.

Answer: 160 in.^{2}

Explanation: No, I disagree with Jose as he found surface area instead of the lateral area, so the lateral area is

4×1/2×10×8

= 2×10×8

= 160 in.^{2}

### Surface Area of Pyramids – Page No. 619

**Use a net to find the surface area of the square pyramid.**

Question 1.

_______ mm^{2}

Answer: 95 mm^{2}

Explanation:

The area of the base is 5×5= 25 mm^{2}

The area of one face is 1/2×5×7

= 35/2

= 17.5 mm^{2}

The surface area is 25+4×17.5

= 25+4×17.5

= 25+70

= 95 mm^{2}

Question 2.

_______ cm^{2}

Answer: 612 cm^{2}

Explanation:

The area of the base is 18×18= 324 cm^{2}

The area of one face is 1/2×18×8

= 18×4

= 72 cm^{2}

The surface area is 324+4×72

= 25+4×17.5

= 25+70

= 612 cm^{2}

Question 3.

_______ yd^{2}

Answer: 51.25 yd^{2}

Explanation:

The area of the base is 2.5×2.5= 6.25 mm^{2}

The area of one face is 1/2×2.5×9

= 22.5/2

= 11.25 yd^{2}

The surface area is 25+4×17.5

= 6.25+4×11.25

= 6.25+45

= 51.25 yd^{2}

Question 4.

_______ in.^{2}

Answer: 180 in^{2}

Explanation:

The area of the base is 10×10= 100 in^{2}

The area of one face is 1/2×4×10

= 2×10

= 20 in^{2}

The surface area is 100+4×20

= 100+4×20

= 100+80

= 180 in^{2}

**Problem Solving**

Question 5.

Cho is building a sandcastle in the shape of a triangular pyramid. The area of the base is 7 square feet. Each side of the base has a length of 4 feet and the height of each face is 2 feet. What is the surface area of the pyramid?

_______ ft^{2}

Answer: 19 ft^{2}

Explanation:

The area of one face is 1/2×4×2= 4 ft^{2}

The surface area of the triangular pyramid is 7+3×4

= 7+12

= 19 ft^{2}

Question 7.

Write and solve a problem finding the lateral area of an object shaped like a square pyramid.

Answer: Mary has a triangular pyramid with a base of 10cm and a height of 15cm. What is the lateral area of the pyramid?

Explanation:

The area of one face is 1/2×10×15

= 5×15

= 75 cm^{2}

The lateral area of the triangular pyramid is 3×75

= 225 cm^{2}

### Lesson Check – Page No. 620

Question 1.

A square pyramid has a base with a side length of 12 in. Each face has a height of 7 in. What is the surface area of the pyramid?

_______ in.^{2}

Answer: 312 in.^{2}

Explanation:

The area of the base is 12×12= 144 in.^{2}

The area of one face is 1/2×12×7

= 6×7

= 42 in.^{2}

The surface area of the square pyramid is 144+4×42

= 144+ 168

= 312 in.^{2}

**Spiral Review**

Question 3.

What is the linear equation represented by the graph?

Answer: y=x+1.

Explanation: As the figure represents that every y value is 1 more than the corresponding x value, so the linear equation is y=x+1.

Question 4.

A regular octagon has sides measuring about 4 cm. If the octagon is divided into 8 congruent triangles, each has a height of 5 cm. What is the area of the octagon?

_______ cm^{2}

Answer:

Explanation:

Area is 1/2bh

= 1/2× 4×5

= 2×5

= 10 cm^{2}

So the area of each triangle is 10 cm^{2}

and the area of the octagon is 8×10= 80 cm^{2}

Question 5.

Carly draws quadrilateral JKLM with vertices J(−3, 3), K(3, 3), L(2, −1), and M(−2, −1). What is the best way to classify the quadrilateral?

Answer: It is a Trapezoid.

Explanation: It is a Trapezoid.

### Mid-Chapter Checkpoint – Vocabulary – Page No. 621

**Choose the best term from the box to complete the sentence.**

Question 1.

_____ is the sum of the areas of all the faces, or surfaces, of a solid figure.

Answer: Surface area is the sum of the areas of all the faces, or surfaces, of a solid figure.

Question 2.

A three-dimensional figure having length, width, and height is called a(n) _____.

Answer: A three-dimensional figure having length, width, and height is called a(n) solid figure.

Question 3.

The _____ of a solid figure is the sum of the areas of its lateral faces.

Answer: The lateral area of a solid figure is the sum of the areas of its lateral faces.

**Concepts and Skills**

Question 4.

Identify and draw a net for the solid figure.

Answer: Triangular prism

Explanation:

Question 5.

Use a net to find the lateral area of the square pyramid.

_______ in.^{2}

Answer: 216 in.^{2}

Explanation:

The area of one face is 1/2×9×12

= 9×6

= 54 in.^{2}

The lateral area of the square pyramid is 4×54= 216 in.^{2}

Question 6.

Use a net to find the surface area of the prism.

_______ cm^{2}

Answer: 310 cm^{2}

Explanation:

The area of face A and E is 10×5= 50 cm^{2}

The area of face B and F is 10×7= 70 cm^{2}

The area of face C and D is 7×5= 35 cm^{2}

The surface area of the prism is 2×50+2×70+2×35

= 100+140+70

= 310 cm^{2}

### Page No. 622

Question 7.

A machine cuts nets from flat pieces of cardboard. The nets can be folded into triangular pyramids and used as pieces in a board game. What shapes appear in the net? How many of each shape are there?

Answer: 4 triangles.

Explanation: There are 4 triangles.

Question 8.

Fran’s filing cabinet is 6 feet tall, 1 \(\frac{1}{3}\) feet wide, and 3 feet deep. She plans to paint all sides except the bottom of the cabinet. Find the area of the sides she intends to paint.

_______ ft^{2}

Answer: 56 ft^{2}

Explanation:

The two lateral face area is 6×1 1/3

= 6× 4/3

= 2×4

= 8 ft^{2}

The area of the other two lateral faces is 6×3= 18

The area of the top and bottom is 3× 1 1/3

= 3× 4/3

= 4 ft^{2}

The area of the sides she intends to paint is 2×8+2×18+4

= 16+36+4

= 56 ft^{2}

Question 10.

What is the surface area of a storage box that measures 15 centimeters by 12 centimeters by 10 centimeters?

_______ cm^{2}

Answer: 900 cm^{2}

Explanation:

The area of two faces is 15×12= 180 cm^{2}

The area of another two faces is 15×10= 150 cm^{2}

The area of the other two faces is 10×12= 120 cm^{2}

So surface area of the storage box is 2×180+2×150+2×120 cm^{2}

= 360+300+240

= 900 cm^{2}

Question 11.

A small refrigerator is a cube with a side length of 16 inches. Use the formula S = 6s^{2} to find the surface area of the cube.

_______ in.^{2}

Answer: 1,536 in.^{2}

Explanation:

Area = s^{2}

= 6×(16)^{2}

= 6× 256

= 1,536 in.^{2}

### Share and Show – Page No. 625

Question 1.

A prism is filled with 38 cubes with a side length of \(\frac{1}{2}\) unit. What is the volume of the prism in cubic units?

_______ \(\frac{□}{□}\) cubic units

Answer: 4.75 cubic units

Explanation:

The volume of the cube is S^{3
}The volume of a cube with S= (1/2)^{3
}= 1/2×1/2×1/2

= 1/8

= 0.125 cubic units

As there are 38 cubes so 38×0.125= 4.75 cubic units.

**Find the volume of the rectangular prism.**

Question 3.

_______ cubic units

Answer: 33 cubic units.

Explanation:

The volume of the rectangular prism is= Width×Height×Length

= 5 1/2 ×3×2

= 11/2 ×3×2

= 33 cubic units.

Question 4.

_______ \(\frac{□}{□}\) cubic units

Answer: 91 1/8 cubic units.

Explanation:

The volume of the rectangular prism is= Width×Height×Length

= 4 1/2 ×4 1/2×4 1/2

= 9/2 ×9/2×9/2

= 729/8

= 91 1/8 cubic units.

Question 5.

Theodore wants to put three flowering plants in his window box. The window box is shaped like a rectangular prism that is 30.5 in. long, 6 in. wide, and 6 in. deep. The three plants need a total of 1,200 in.^{3} of potting soil to grow well. Is the box large enough? Explain.

Answer: No, the box is not large enough as the three plants need a total of 1,200 in.^{3} and here volume is 1,098 in.^{3}

Explanation:

Volume= Width×Height×Length

= 30.5×6×6

= 1,098 in.^{3}

### Problem Solving + Applications – Page No. 626

**Use the diagram for 7–10.**

Question 7.

Karyn is using a set of building blocks shaped like rectangular prisms to make a model. The three types of blocks she has are shown at right. What is the volume of an A block? (Do not include the pegs on top.)

\(\frac{□}{□}\) cubic units

Answer: 1/2 cubic units

Explanation: Volume = Width×Height×Length

= 1× 1/2 ×1

= 1/2 cubic units

Question 8.

How many A blocks would you need to take up the same amount of space as a C block?

_______ A blocks

Answer: No of blocks required to take up the same amount of space as a C block is 4 A blocks.

Explanation: Volume = Width×Height×Length

= 1×2×1

= 2 cubic unit

No of blocks required to take up the same amount of space as a C block is 1/2 ÷2

= 2×2

= 4 A blocks

Question 9.

Karyn puts a B block, two C blocks, and three A blocks together. What is the total volume of these blocks?

_______ \(\frac{□}{□}\) cubic units

Answer: 6 1/2 cubic units

Explanation: The volume of A block is

Volume = Width×Height×Length

= 1×1 ×1/2

= 1/2 cubic units.

As Karyn puts three A blocks together, so 3× 1/2= 3/2 cubic units.

The volume of B block is

Volume = Width×Height×Length

= 1×1 × 1

= 1 cubic units.

As Karyn puts only one B, so 1 cubic unit.

The volume of C block is

Volume = Width×Height×Length

= 2×1×1

= 2 cubic units.

As Karyn puts two C blocks together, so 2× 2= 4 cubic units.

So, the total volume of these blocks is 3/2 + 1+ 4

= 3/2+5

= 13/2

= 6 1/2 cubic units

Question 11.

Verify the Reasoning of Others Jo says that you can use V = l × w × h or V = h × w × l to find the volume of a rectangular prism. Does Jo’s statement make sense? Explain.

Answer: Yes

Explanation: Yes, Jo’s statement makes sense because by the commutative property we can change the order of the variables of length, width, height and both will produce the same result.

Question 12.

A box measures 5 units by 3 units by 2 \(\frac{1}{2}\) units. For numbers 12a–12b, select True or False for the statement.

12a. The greatest number of cubes with a side length of \(\frac{1}{2}\) unit that can be packed inside the box is 300.

12b. The volume of the box is 37 \(\frac{1}{2}\) cubic units.

12a. __________

12b. __________

Answer:

12a True.

12b True.

Explanation: The volume of the cube is S^{3
}The volume of a cube with S= (1/2)^{3
}= 1/2×1/2×1/2

= 1/8 cubic units

As there are 300 cubes so 300× 1/8= 75/2

= 37 1/2 cubic units.

### Fractions and Volume – Page No. 627

**Find the volume of the rectangular prism.**

Question 1.

_______ \(\frac{□}{□}\) cubic units

Answer: 6 3/4 cubic units

Explanation: Volume = Width×Height×Length

= 3× 1 1/2× 1 1/2

= 3× 3/2 × 3/2

= 27/4

= 6 3/4 cubic units

Question 2.

_______ \(\frac{□}{□}\) cubic units

Answer: 22 1/2 cubic units

Explanation: Volume = Width×Height×Length

= 5×1× 4 1/2

= 5× 9/2

= 45/2

= 22 1/2 cubic units

Question 3.

_______ \(\frac{□}{□}\) cubic units

Answer: 16 1/2 cubic units.

Explanation: Volume = Width×Height×Length

= 5 1/2× 1 1/2× 2

= 11/2×3/2×2

= 33/2

= 16 1/2 cubic units.

Question 4.

_______ \(\frac{□}{□}\) cubic units

Answer: 28 1/8 cubic units.

Explanation: Volume = Width×Height×Length

= 2 1/2× 2 1/2 × 4 1/2

= 5/2 × 5/2 × 9/2

= 225/8

= 28 1/8 cubic units.

**Problem Solving**

Question 6.

A shipping crate is shaped like a rectangular prism. It is 5 \(\frac{1}{2}\) feet long by 3 feet wide by 3 feet high. What is the volume of the crate?

_______ \(\frac{□}{□}\) ft^{3}

Answer: 49 1/2 ft^{3}

Explanation: Volume = Width×Height×Length

= 5 1/2 × 3 × 3

= 11/2 ×9

= 99/2

= 49 1/2 ft^{3}

Question 7.

How many cubes with a side length of \(\frac{1}{4}\) unit would it take to make a unit cube? Explain how you determined your answer.

Answer: There will be 4×4×4= 64 cubes and 1/4 unit in the unit cube.

Explanation:

As the unit cube has a 1 unit length, 1 unit wide, and 1 unit height

So length 4 cubes = 4× 1/4= 1 unit

width 4 cubes = 4× 1/4= 1 unit

height 4 cubes = 4× 1/4= 1 unit

So there will be 4×4×4= 64 cubes and 1/4 unit in the unit cube.

### Lesson Check – Page No. 628

Question 1.

A rectangular prism is 4 units by 2 \(\frac{1}{2}\) units by 1 \(\frac{1}{2}\) units. How many cubes with a side length of \(\frac{1}{2}\) unit will completely fill the prism?

Answer: 120 cubes

Explanation:

No of cubes with a side length of 1/2 unit is

Length 8 cubes= 8× 1/2= 4 units

Width 5 cubes= 5× 1/2= 5/2= 2 1/2 units

Height 3 cubes= 3× 1/2= 3/2= 1 1/2 units

So there are 8×5×3= 120 cubes in the prism.

Question 2.

A rectangular prism is filled with 196 cubes with \(\frac{1}{2}\)-unit side lengths. What is the volume of the prism in cubic units?

_______ \(\frac{□}{□}\) cubic units

Answer: 24 1/2 cubic units.

Explanation: It takes 8 cubes with a side length of 1/2 to form a unit cube, so the volume of the prism in the cubic units is 196÷8= 24 1/2 cubic units.

**Spiral Review**

Question 4.

A flag for the sports club is a rectangle measuring 20 inches by 32 inches. Within the rectangle is a yellow square with a side length of 6 inches. What is the area of the flag that is not part of the yellow square?

_______ in.^{2}

Answer: 604 in.^{2}

Explanation: Area of a flag= Length×width

= 20×32

= 640 in.^{2}

Area of the yellow square= S^{2}

= 6

= 36 in.^{2}

So the area of the flag that is not a part of the yellow square is 640-36= 604 in.^{2}

Question 5.

What is the surface area of the rectangular prism shown by the net?

_______ square units

Answer: 80 square units

Explanation:

Area of two faces is 12 squares

Area of other two faces is 16 squares

Area of another two faces is 12 squares

So the surface area is 2×12+2×16+2×12

= 24+32+24

= 80 square units

Question 6.

What is the surface area of the square pyramid?

_______ cm^{2}

Answer: 161 cm^{2}

Explanation: The area of the base is 7×7= 49 cm^{2}

And the area of one face is 1/2 × 7× 8

= 7×4

= 28 cm^{2}

The surface area of the square pyramid is 49+4×28

= 49+112

= 161 cm^{2}

### Share and Show – Page No. 631

**Find the volume.**

Question 1.

_______ \(\frac{□}{□}\) in.^{3}

Answer: 3,937 1/2 in.^{3}

Explanation: Volume= Length× wide× heght

= 10 1/2 ×15 × 25

= 11/2 × 15 × 25

= 4,125/2

= 3,937 1/2 in.^{3}

Question 2.

_______ \(\frac{□}{□}\) in.^{3}

Answer: 27/512 in.^{3}

Explanation: Volume= Length× wide× height

=3/8 ×3/8 × 3/8

= 27/512 in.^{3}

**On Your Own**

**Find the volume of the prism.**

Question 3.

_______ \(\frac{□}{□}\) in.^{3}

Answer: 690 5/8in.^{3}

Explanation: Volume= Length× wide× height

= 8 1/2 × 6 1/2 × 12 1/2

= 17/2 × 13/2× 25/2

= 5525/2

= 690 5/8in.^{3}

Question 4.

_______ \(\frac{□}{□}\) in.^{3}

Answer: 125/4096 in.^{3}

Explanation: Volume= Length× wide× height

= 5/16 ×5/16 × 5/16

= 125/4096 in.^{3}

Question 5.

_______ yd^{3}

Answer: 20 yd^{3}

Explanation:

Area= 3 1/3 yd^{2}

So Area= wide×height

3 1/3= w × 1 1/3

10/3= w× 4/3

w= 10/3 × 3/4

w= 5/2

w= 2.5 yd

Volume= Length×width×height

= 6× 2.5× 1 1/3

= 6×2.5× 4/3

= 2×2.5×4

= 20 yd^{3}

Question 7.

Abraham has a toy box that is in the shape of a rectangular prism.

The volume is _____.

_______ \(\frac{□}{□}\) ft^{3}

Answer: 33 3/4 ft^{3}

Explanation: Volume of rectangular prism is= Length×width×height

= 4 1/2× 2 1/2× 3

= 9/2 × 5/2× 3

= 135/3

= 33 3/4 ft^{3}

### Aquariums – Page No. 632

Large public aquariums like the Tennessee Aquarium in Chattanooga have a wide variety of freshwater and saltwater fish species from around the world. The fish are kept in tanks of various sizes.

The table shows information about several tanks in the aquarium. Each tank is a rectangular prism.

Find the length of Tank 1.

V = l w h

52,500 = l × 30 × 35

\(\frac{52,500}{1,050}\) = l

50 = l

So, the length of Tank 1 is 50 cm.

**Solve.**

Question 8.

Find the width of Tank 2 and the height of Tank 3.

Answer: Width of Tank 2= 8m, Height of the Tank 3= 10 m

Explanation:

The volume of Tank 2= 384 m^{3}

so V= LWH

384= 12×W×4

W= 384/48

W= 8 m

So the width of Tank 2= 8m

The volume of Tank 3= 2160 m

So V= LWH

2160= 18×12×H

H= 2160/216

H= 10 m

So the height of the Tank 3= 10 m

Question 9.

To keep the fish healthy, there should be a correct ratio of water to fish in the tank. One recommended ratio is 9 L of water for every 2 fish. Find the volume of Tank 4. Then use the equivalencies 1 cm^{3} = 1 mL and 1,000 mL = 1 L to find how many fish can be safely kept in Tank 4.

Answer: 35 Fishes

Explanation:

The volume of Tank 4 = LWH

= 72×55×40

= 1,58,400 cm^{3}

As 1 cm^{3} = 1 mL and 1,000 mL = 1 L

1,58,400 cm^{3} = 1,58,400 mL and 1,58,400 mL = 158.4 L

So tank can keep safely (158.4÷ 9)×2

= (17.6)× 2 = 35.2

= 35 Fishes

Question 10.

Use Reasoning Give another set of dimensions for a tank that would have the same volume as Tank 2. Explain how you found your answer.

Answer: Another set of dimensions for a tank that would have the same volume as Tank 2 is 8m by 8m by 6m.

So when we multiply the product will be 384

### Volume of Rectangular Prisms – Page No. 633

**Find the volume.**

Question 1.

_______ \(\frac{□}{□}\) m^{3}

Answer: 150 5/16 m^{3}

Explanation: Volume= Length×width×height

= 5× 3 1/4× 9 1/4

= 5× 13/4 × 37/4

= 2405/16

= 150 5/16 m^{3}

Question 2.

_______ \(\frac{□}{□}\) in.^{3}

Answer: 27 1/2 in.^{3}

Explanation: Volume= Length×width×height

= 5 1/2 × 2 1/2 × 2

= 11/2 × 5/2 × 2

= 55/2

= 27 1/2 in.^{3}

Question 3.

_______ \(\frac{□}{□}\) mm^{3}

Answer: 91 1/8 mm^{3}

Explanation: Volume= Length×width×height

= 4 1/2 × 4 1/2 × 4 1/2

= 9/2 × 9/2 × 9/2

= 729/8

= 91 1/8 mm^{3}

Question 4.

_______ \(\frac{□}{□}\) ft^{3}

Answer: 112 1/2 ft^{3}

Explanation: Volume= Length×width×height

= 7 1/2 × 2 1/2 × 6

= 15/2 × 5/2 × 6

= 225/2

= 112 1/2 ft^{3}

Question 5.

_______ m^{3}

Answer: 36 m^{3}

Explanation:

The area of shaded face is Length × width= 8 m^{2}

Volume of the prism= Length×width×height

= 8 × 4 1/2

= 8 × 9/2

= 4 × 9

= 36 m^{3}

Question 6.

_______ \(\frac{□}{□}\) ft^{3
}

Answer: 30 3/8 ft^{3}

Explanation: Volume of the prism= Length×width×height

= 2 1/4 × 6 × 2 1/4

= 9/4 × 6 × 9/4

= 243/8

= 30 3/8 ft^{3}

**Problem Solving**

Question 9.

Explain how you can find the side length of a rectangular prism if you are given the volume and the two other measurements. Does this process change if one of the measurements includes a fraction?

Answer: We can find the side length of a rectangular prism if we are given the volume and the two other measurements by dividing the value of the volume by the product of the values of the width and height of the prism. And the process doesn’t change if one of the measurements includes a fraction.

### Lesson Check – Page No. 634

Question 1.

A kitchen sink is a rectangular prism with a length of 19 \(\frac{7}{8}\) inches, a width of 14 \(\frac{3}{4}\) inches, and height of 10 inches. Estimate the volume of the sink.

Answer: 3,000 in.^{3}

Explanation: Length = 19 7/8 as the number was close to 20 and width 14 3/4 which is close to 15 and height is 10

So Volume= LBH

= 20 × 15 × 10

= 3,000 in.^{3}

**Spiral Review**

Question 3.

Carrie started at the southeast corner of Franklin Park, walked north 240 yards, turned and walked west 80 yards, and then turned and walked diagonally back to where she started. What is the area of the triangle enclosed by the path she walked?

_______ yd^{2}

Answer: 9,600 yd^{2}

Explanation:

Area of triangle= 1/2 bh

= 1/2 × 240 × 80

= 240 × 40

= 9,600 yd^{2}

Question 4.

The dimensions of a rectangular garage are 100 times the dimensions of a floor plan of the garage. The area of the floor plan is 8 square inches. What is the area of the garage?

Answer: 80,000 in^{2}

Explanation: As 1 in^{2}= 10,000 in^{2}, so area of the floor plan 8 in

= 8×10000

= 80,000 in^{2}

Question 5.

Shiloh wants to create a paper-mâché box shaped like a rectangular prism. If the box will be 4 inches by 5 inches by 8 inches, how much paper does she need to cover the box?

Answer: 184 in^{2}

Explanation: Area of the rectangular prism= 2(wl+hl+hw)

= 2(4×5 + 5×8 + 8×4)

= 2(20+40+32)

= 2(92)

= 184 in^{2}

Question 6.

A box is filled with 220 cubes with a side length of \(\frac{1}{2}\) unit. What is the volume of the box in cubic units?

_______ \(\frac{□}{□}\) cubic units

Answer: 27.5 cubic units.

Explanation: The volume of a cube side is (1/2)^{3} = 1/8

So 220 cubes= 220× 1/8

= 27.5 cubic units.

### Share and Show – Page No. 637

Question 1.

An aquarium tank in the shape of a rectangular prism is 60 cm long, 30 cm wide, and 24 cm high. The top of the tank is open, and the glass used to make the tank is 1 cm thick. How much water can the tank hold?

_______ cm^{3}

Answer: So tank can hold 37,352 cm^{3}

Explanation: As Volume= LBH

Let’s find the inner dimensions of the tank, so 60-2 × 30-2 × 24-1

= 58×28×23

= 37,352 cm^{3}

Question 2.

What if, to provide greater strength, the glass bottom were increased to a thickness of 4 cm? How much less water would the tank hold?

_______ cm^{3}

Answer: 4,872 cm^{3}

Explanation: As the glass bottom was increased to a thickness of 4 cm, 60-2 × 30-2 × 24-4

= 58×28×20

= 32,480 cm^{3}

So the tank can hold 37,352- 32,480= 4,872 cm^{3}

Question 4.

The Louvre Museum in Paris, France, has a square pyramid made of glass in its central courtyard. The four triangular faces of the pyramid have bases of 35 meters and heights of 27.8 meters. What is the area of glass used for the four triangular faces of the pyramid?

Answer: 1946 m^{2}

Explanation: The area of one face is 1/2 × 35 × 27.8= 486.5 m^{2}

And the area of glass used for the four triangular faces of the pyramid is 4×486.5= 1946 m^{2}

### On Your Own – Page No. 638

Question 5.

A rectangular prism-shaped block of wood measures 3 m by 1 \(\frac{1}{2}\) m by 1 \(\frac{1}{2}\) m. How much of the block must a carpenter carve away to obtain a prism that measures 2 m by \(\frac{1}{2}\) m by \(\frac{1}{2}\) m?

_______ \(\frac{□}{□}\) m^{3}

Answer: 6 1/4 m^{3}

Explanation: The volume of the original block= LWH

= 3 × 1 1/2 × 1 1/2

= 3× 3/2 × 3/2

= 27/4

= 6 3/4 m^{2}

And volume of carpenter carve is 2× 1/2 × 1/2

= 1/2 m^{2}

So, the carpenter must carve 27/4 – 1/2

= 25/2

= 6 1/4 m^{3}

Question 6.

The carpenter (Problem 5) varnished the outside of the smaller piece of wood, all except for the bottom, which measures \(\frac{1}{2}\) m by \(\frac{1}{2}\) m. Varnish costs $2.00 per square meter. What was the cost of varnishing the wood?

$ _______

Answer: $8.50

Explanation: The area of two lateral faces are 2×1/2= 1 m^{2}

The area of the other two lateral faces are 2×1/2= 1 m^{2}

The area of the top and bottom is 1/2×1/2= 1/4 m^{2}

And the surface area is 2×1 + 2×1 + 1/4

= 2+2+1/4

= 17/4

= 4.25 m^{2}

And the cost of vanishing the wood is $2.00× 4.25= $8.50

Question 8.

Describe A rectangular prism-shaped box that measures 6 cm by 5 cm by 4 cm. A cube-shaped box has a side length of 2 cm. How many of the cube-shaped boxes will fit into the rectangular prism-shaped box? Describe how you found your answer.

Answer: 12 cube-shaped boxes

Explanation: As 6 small boxes can fit on the base i.e 6 cm by 5 cm, as height is 4cm there can be a second layer of 6 small boxes. So, there will be a total of 12 cube-shaped boxes and will fit into a rectangular prism-shaped box

Question 9.

Justin is covering the outside of an open shoe box with colorful paper for a class project. The shoe box is 30 cm long, 20 cm wide, and 13 cm high. How many square centimeters of paper are needed to cover the outside of the open shoe box? Explain your strategy

_______ cm^{2}

Answer: 1,900 cm^{2}

Explanation:

The area of the two lateral faces of the shoebox is 20×13= 260 cm^{2}

The area of another two lateral faces of the shoebox is 30×13= 390 cm^{2}

The area of the top and bottom is 30×20= 600 cm^{2}

So, the surface area of the shoebox without the top is 2×260 + 2× 390 + 600

= 520+780+600

= 1,900 cm^{2}

### Problem Solving Geometric Measurements – Page No. 639

**Read each problem and solve.**

Question 1.

The outside of an aquarium tank is 50 cm long, 50 cm wide, and 30 cm high. It is open at the top. The glass used to make the tank is 1 cm thick. How much water can the tank hold?

_______ cm^{3}

Answer: So water tank can hold 66,816 cm^{3}

Explanation: The volume of inner dimensions of the aquarium is 50-2 × 50-2 × 30-1

= 48×48×29

= 66,816 cm^{3}

So water tank can hold 66,816 cm^{3}

Question 2.

Arnie keeps his pet snake in an open-topped glass cage. The outside of the cage is 73 cm long, 60 cm wide, and 38 cm high. The glass used to make the cage is 0.5 cm thick. What is the inside volume of the cage?

_______ cm^{3}

Answer: The volume of the cage is 1,59,300 cm^{3}

Explanation: The volume of inner dimensions is 73-1 × 60-1 × 38-0.5

= 72×59×37.5

= 1,59,300 cm^{3
}So, the volume of the cage is 1,59,300 cm^{3}

Question 3.

A display number cube measures 20 in. on a side. The sides are numbered 1–6. The odd-numbered sides are covered in blue fabric and the even-numbered sides are covered in red fabric. How much red fabric was used?

_______ in.^{2}

Answer: 1200 in.^{2}

Explanation: The area of each side of a cube is 20×20= 400 in.^{2}, as there are 3 even-numbered sides on the cube. So there will be

3×400= 1200 in.^{2}

Question 4.

The caps on the tops of staircase posts are shaped like square pyramids. The side length of the base of each cap is 4 inches. The height of the face of each cap is 5 inches. What is the surface area of the caps for two posts?

_______ in.^{2}

Answer: 112 in.^{2}

Explanation: The area of the base is 4×4= 16 in.^{2}

The area of one face is 1/2×5×4= 10 in.^{2}

The surface area of one cap is 16+4×10

= 16+40

= 56 in.^{2}

And the surface area of the caps for two posts is 2×56= 112 in.^{2}

Question 6.

Write and solve a problem for which you use part of the formula for the surface area of a triangular prism.

Answer: In a triangular prism, the triangular end has a base of 5cm and a height is 8 cm. The length of each side is 4 cm and the height of the prism is 10 cm. What is the lateral area of this triangular prism?

Explanation: The area of two triangular faces is 1/2 × 5 × 8

= 5×4

= 20 cm^{2}

The area of two rectangular faces is 4×10= 40 cm^{2}

The lateral area is 2×20+2×40

= 40+80

= 120 cm^{2}

### Lesson Check – Page No. 640

Question 1.

Maria wants to know how much wax she will need to fill a candle mold shaped like a rectangular prism. What measure should she find?

Answer: Maria needs to find the volume of the mold.

Question 2.

The outside of a closed glass display case measures 22 inches by 15 inches by \(\frac{1}{2}\) inches. The glass is 12 inch thick. How much air is contained in the case?

_______ in.^{3}

Answer: 3381 in.^{3}

Explanation: The inner dimensions are 22-1× 15-1 × 12- 1/2

= 21 ×14×23/2

= 3381 in.^{3}

**Spiral Review**

Question 4.

Sierra has plotted two vertices of a rectangle at (3, 2) and (8, 2). What is the length of the side of the rectangle?

_______ units

Answer: 5 units.

Explanation: The length of the side of the rectangle is 8-3= 5 units.

Question 5.

What is the surface area of the square pyramid?

_______ m^{2}

Answer: 104 m^{2}

Explanation: The area of the base 4×4= 16

The area of the one face is 1/2 × 4 × 11

= 2×11

= 22 m^{2}

The surface area of the square pyramid is 16+4×22

= 16+88

= 104 m^{2}

Question 6.

A shipping company has a rule that all packages must be rectangular prisms with a volume of no more than 9 cubic feet. What is the maximum measure for the height of a box that has a width of 1.5 feet and a length of 3 feet?

_______ feet

Answer: 2 feet.

Explanation: As given volume = 9 cubic feet

So 1.5×3×H < 9

4.5×H < 9

H< 9/4.5

and H<2

So maximum measure for the height of the box is 2 feet.

### Chapter 11 Review/Test – Page No. 641

Question 1.

Elaine makes a rectangular pyramid from paper.

The base is a _____. The lateral faces are _____.

The base is a ___________ .

The lateral faces are ___________ .

Answer:

The base is a rectangle.

The lateral faces are triangles.

Question 2.

Darrell paints all sides except the bottom of the box shown below.

Select the expressions that show how to find the surface area that Darrell painted. Mark all that apply.

Options:

a. 240 + 240 + 180 + 180 + 300 + 300

b. 2(20 × 12) + 2(15 × 12) + (20 × 15)

c. (20 × 12) + (20 × 12) + (15 × 12) + (15 × 12) + (20 × 15)

d. 20 × 15 × 12

Answer: b,c

Explanation: The expressions that show how to find the surface area is 2(20 × 12) + 2(15 × 12) + (20 × 15), (20 × 12) + (20 × 12) + (15 × 12) + (15 × 12) + (20 × 15)

Question 4.

A triangular pyramid has a base with an area of 11.3 square meters, and lateral faces with bases of 5.1 meters and heights of 9 meters. Write an expression that can be used to find the surface area of the triangular pyramid.

Answer: 11.3+ 3 × 1/2+ 5.1×9

Explanation: The expression that can be used to find the surface area of the triangular pyramid is 11.3+ 3 × 1/2+ 5.1×9

### Page No. 642

Question 5.

Jeremy makes a paperweight for his mother in the shape of a square pyramid. The base of the pyramid has a side length of 4 centimeters, and the lateral faces have heights of 5 centimeters. After he finishes, he realizes that the paperweight is too small and decides to make another one. To make the second pyramid, he doubles the length of the base in the first pyramid.

For numbers 5a–5c, choose Yes or No to indicate whether the statement is correct.

5a. The surface area of the second pyramid is 144 cm^{2}.

5b. The surface area doubled from the first pyramid to the second pyramid.

5c. The lateral area doubled from the first pyramid to the second pyramid.

5a. ___________

5b. ___________

5c. ___________

Answer:

5a. True.

5b. False

5c. True.

Explanation:

The area of the base is 4×4= 16 cm^{2}.

The area of one face is 1/2×4×5

= 2×5

= 10 cm^{2}.

The surface area of the First pyramid is 16+ 4×10

= 16+40

= 56 cm^{2}.

The area of the base is 8×8= 64

The area of one face is 1/2×8×5

= 4×5

= 20 cm^{2}.

The surface area od the second pyramid is 64+ 4×20

= 64+80

= 144 cm^{2}.

Question 6.

Identify the figure shown and find its surface area. Explain how you found your answer.

Answer: 369 in^{2}

Explanation:

The area of the base is 9×9= 81 in^{2}

The area of one face is 1//2 × 16× 9

= 8×9

= 72 in^{2}

The surface area of a square pyramid is 81+ 4× 72

= 81+ 288

= 369 in^{2}

Question 7.

Dominique has a box of sewing buttons that is in the shape of a rectangular prism.

The volume of the box is 2 \(\frac{1}{2}\) in. × 3 \(\frac{1}{2}\) in. × _____ = _____.

Answer: 17.5 in^{3}

Explanation: The volume of the box is 2 1/2 × 3 1/2 × 2

= 5/2 × 7/2 × 2

= 5/2 × 7

= 35/2

= 17.5 in^{3}

### Page No. 643

Question 8.

Emily has a decorative box that is shaped like a cube with a height of 5 inches. What is the surface area of the box?

_______ in.^{2}

Answer: 150 in.^{2}

Explanation: Surface area of the box is 6 a^{2}

So 6 × 5^{2}

= 6×5×5^{2}

= 150 in.^{2}

Question 9.

Albert recently purchased a fish tank for his home. Match each question with the geometric measure that would be most appropriate for each scenario.

Answer:

Question 10.

Select the expressions that show the volume of the rectangular prism. Mark all that apply.

Options:

a. 2(2 units × 2 \(\frac{1}{2 }\) units) + 2(2 units × \(\frac{1}{2}\) unit) + 2(\(\frac{1}{2}\) unit × 2 \(\frac{1}{2}\) units)

b. 2(2 units × \(\frac{1}{2}\) unit) + 4(2 units × 2 \(\frac{1}{2}\) units)

c. 2 units × \(\frac{1}{2}\) unit × 2 \(\frac{1}{2}\) units

d. 2.5 cubic units

Answer: c, d

Explanation: 2 units ×1/2 unit × 2 1/2 units and 2.5 cubic units

### Page No. 644

Question 11.

For numbers 11a–11d, select True or False for the statement.

11a. The area of face A is 8 square units.

11b. The area of face B is 10 square units.

11c. The area of face C is 8 square units.

11d. The surface area of the prism is 56 square units.

11a. ___________

11b. ___________

11c. ___________

11d. ___________

Answer:

11a. True.

11b. True.

11c. False.

11d. False.

Explanation:

The area of the face A is 4×2= 8 square units

The area of the face B is 5×2= 10 square units

The area of the face C is 5×4= 20 square units

So the surface area is 2×8+2×10+2×20

= 16+20+40

= 76 square units

Question 12.

Stella received a package in the shape of a rectangular prism. The box has a length of 2 \(\frac{1}{2}\) feet, a width of 1 \(\frac{1}{2}\) feet, and a height of 4 feet.

Part A

Stella wants to cover the box with wrapping paper. How much paper will she need? Explain how you found your answer

Answer: 39.5 ft^{2}

Explanation:

The area of two lateral faces is 4 × 2 1/2

= 4 × 5/2

= 2×5

= 10 ft^{2}

The area of another two lateral faces is 4 × 1 1/2

= 4 × 3/2

= 2×3

= 6 ft^{2}

The area of the top and bottom is 2 1/2 × 1 1/2

= 5/2 × 3/2

= 15/4

= 3 3/4 ft^{2}

So Stella need 2×10+ 2×6 + 2 × 15/4

= 20+ 12+15/2

= 20+12+7.5

= 39.5 ft^{2}

Question 12.

Part B

Can the box hold 16 cubic feet of packing peanuts? Explain how you know

Answer: The box cannot hold 16 cubic feet of the packing peanuts

Explanation: Volume = LWH

= 2 1/2 ×1 1/2 × 4

= 5/2 × 3/2 ×4

= 5×3

= 15 ft^{3
}So the box cannot hold 16 cubic feet of packing peanuts.

### Page No. 645

Question 13.

A box measures 6 units by \(\frac{1}{2}\) unit by 2 \(\frac{1}{2}\) units.

For numbers 13a–13b, select True or False for the statement.

13a. The greatest number of cubes with a side length of \(\frac{1}{2}\) unit that can be packed inside the box is 60.

13b. The volume of the box is 7 \(\frac{1}{2}\) cubic units.

13a. ___________

13b. ___________

Answer:

13a. True

13b. True.

Explanation:

Length is 12 × 1/2= 6 units

Width is 1× 1/2= 1/2 units

Height is 5× 1/2= 5/2 units

So, the greatest number of cubes with a side length of 1/2 unit that can be packed inside the box is 12×1×5= 60

The volume of the cube is S^{3
}The volume of a cube with S= (1/2)^{3
}= 1/2×1/2×1/2

= 1/8

= 0.125 cubic units

As there are 60 cubes so 60×0.125= 7.5 cubic units.

Question 15.

Lourdes is decorating a toy box for her sister. She will use self-adhesive paper to cover all of the exterior sides except for the bottom of the box. The toy box is 4 feet long, 3 feet wide, and 2 feet high. How many square feet of adhesive paper will Lourdes use to cover the box?

_______ ft^{2}

Answer: 40 ft^{2}

Explanation:

The area of two lateral faces is 4×2= 8 ft^{2}

The area of another two lateral faces is 3×2= 6 ft^{2}

The area of the top and bottom is 4×3= 12 ft^{2}

So Lourdes uses to cover the box is 2×8 + 2×6 + 12

= 16+12+12

= 40 ft^{2}

Question 16.

Gary wants to build a shed shaped like a rectangular prism in his backyard. He goes to the store and looks at several different options. The table shows the dimensions and volumes of four different sheds. Use the formula V = l × w × h to complete the table.

Answer:

Length of shed 1= 12 ft

Width of shed 2= 12 ft

Height of shed 3= 6 ft

Volume of shed 4= 1200 ft^{3}

Explanation: Volume= LWH

Volume of shed1= 960 ft

So 960= L×10×8

960= 80×L

L= 960/80

L= 12 ft

Volume of shed2= 2160 ft

So 2160= 18×W×10

960= 180×W

W= 2160/180

W= 12 ft

Volume of shed3= 288 ft

So 288= 12×4×H

288= 48×H

H= 288/48

W= 6 ft

Volume of shed2= 10×12×10

So V= 10×12×10

V= 1200 ft^{3}

### Page No. 646

Question 17.

Tina cut open a cube-shaped microwave box to see the net. How many square faces does this box have?

_______ square faces

Answer: The box has 6 square faces.

Question 18.

Charles is painting a treasure box in the shape of a rectangular prism.

Which nets can be used to represent Charles’ treasure box? Mark all that apply.

Options:

a.

b.

c.

d.

Answer: a and b can be used to represent Charle’s treasure box.

Question 19.

Julianna is lining the inside of a basket with fabric. The basket is in the shape of a rectangular prism that is 29 cm long, 19 cm wide, and 10 cm high. How much fabric is needed to line the inside of the basket if the basket does not have a top? Explain your strategy.

_______ cm^{2}

Answer: 1511 cm^{2}

Explanation: The surface area= 2(WL+HL+HW)

The surface area of the entire basket= 2(19×29)+2(10×29)+2(10×19)

= 2(551)+2(290)+2(190)

= 1102+580+380

= 2,062 cm^{2}

The surface area of the top is 29×19= 551

So Julianna needs 2062-551= 1511 cm^{2}

*Conclusion*

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