## Engage NY Eureka Math 7th Grade Module 3 Lesson 21 Answer Key

### Eureka Math Grade 7 Module 3 Lesson 21 Example Answer Key

Example 1: Lateral Area of a Right Prism

A right triangular prism, a right rectangular prism, and a right pentagonal prism are pictured below, and all have equal heights of h.

a. Write an expression that represents the lateral area of the right triangular prism as the sum of the areas of its lateral faces.

Answer:

a ∙ h + b ∙ h + c ∙ h

b. Write an expression that represents the lateral area of the right rectangular prism as the sum of the areas of its lateral faces.

Answer:

a ∙ h + b ∙ h + a ∙ h + b ∙ h

c. Write an expression that represents the lateral area of the right pentagonal prism as the sum of the areas of its lateral faces.

Answer:

a ∙ h + b ∙ h + c ∙ h + d ∙ h + e ∙ h

d. What value appears often in each expression and why?

Answer:

h; Each prism has a height of h; therefore, each lateral face has a height of h.

e. Rewrite each expression in factored form using the distributive property and the height of each lateral face.

Answer:

h(a + b + c)

h(a + b + a + b)

h(a + b + c + d + e)

f. What do the parentheses in each case represent with respect to the right prisms?

Answer:

The perimeter of the base of the corresponding prism.

g. How can we generalize the lateral area of a right prism into a formula that applies to all right prisms?

Answer:

If LA represents the lateral area of a right prism, P represents the perimeter of the right prism’s base, and h represents the distance between the right prism’s bases, then:

LA = P_{base} ∙ h.

Relevant Vocabulary

RIGHT PRISM: Let E and E’ be two parallel planes. Let B be a triangular or rectangular region or a region that is the union of such regions in the plane E. At each point P of B, consider the segment PP’ perpendicular to E, joining P to a point P’ of the plane E’. The union of all these segments is a solid called a right prism.

There is a region B’ in E’ that is an exact copy of the region B. The regions B and B’ are called the base faces (or just bases) of the prism. The rectangular regions between two corresponding sides of the bases are called lateral faces of the prism. In all, the boundary of a right rectangular prism has 6 faces: 2 base faces and 4 lateral faces. All adjacent faces intersect along segments called edges (base edges and lateral edges).

CUBE: A cube is a right rectangular prism all of whose edges are of equal length.

SURFACE: The surface of a prism is the union of all of its faces (the base faces and lateral faces).

NET: A net is a two-dimensional diagram of the surface of a prism.

1. Why are the lateral faces of right prisms always rectangular regions?

Answer:

Because along a base edge, the line segments PP’ are always perpendicular to the edge, forming a rectangular region.

2. What is the name of the right prism whose bases are rectangles?

Answer:

Right rectangular prism

3. How does this definition of right prism include the interior of the prism?

Answer:

The union of all the line segments fills out the interior.

### Eureka Math Grade 7 Module 3 Lesson 21 Exercise Answer Key

Opening Exercise: Surface Area of a Right Rectangular Prism

On the provided grid, draw a net representing the surfaces of the right rectangular prism (assume each grid line represents 1 inch). Then, find the surface area of the prism by finding the area of the net.

Answer:

There are six rectangular faces that make up the net.

The four rectangles in the center form one long rectangle that is 20 in. by 3 in.

Area = lw

Area = 3 in ∙ 20 in

Area = 60 in^{2}

Two rectangles form the wings, both 6 in by 4 in.

Area = lw

Area = 6 in ∙ 4 in

Area = 24 in^{2}

The area of both wings is 2(24 in^{2} ) = 48 in^{2}.

The total area of the net is

A = 60 in^{2} + 48 in^{2} = 108 in^{2}

The net represents all the surfaces of the rectangular prism, so its area is equal to the surface area of the prism. The surface area of

the right rectangular prism is 108 in^{2}.

Exercise 1.

Marcus thinks that the surface area of the right triangular prism will be half that of the right rectangular prism and wants to use the modified formula SA = \(\frac{1}{2}\) (2lw + 2lh + 2wh). Do you agree or disagree with Marcus? Use nets of the prisms to support your argument.

Answer:

The surface area of the right rectangular prism is 108 in^{2}, so Marcus believes the surface areas of each right triangular prism is 54 in^{2}.

The net of the right triangular prism has one less face than the right rectangular prism. Two of the rectangular faces on the right triangular prism (rectangular regions 1 and 2 in the diagram) are the same faces from the right rectangular prism, so they are the same size. The areas of the triangular bases (triangular regions 3 and 4 in the diagram) are half the area of their corresponding rectangular faces of the right rectangular prism. These four faces of the right triangular prism make up half the surface area of the right rectangular prism before considering the fifth face; no, Marcus is incorrect.

The areas of rectangular faces 1 and 2, plus the areas of the triangular regions 3 and 4 is 54 in^{2}. The last rectangular region has an area of 30 in^{2}. The total area of the net is 54 in^{2} + 30 in^{2} or 84 in^{2}, which is far more than half the surface area of the right rectangular prism.

### Eureka Math Grade 7 Module 3 Lesson 21 Problem Set Answer Key

Question 1.

For each of the following nets, highlight the perimeter of the lateral area, draw the solid represented by the net, indicate the type of solid, and then find the solid’s surface area.

a. Right rectangular prism

Answer:

SA = LA + 2B

LA = P ∙ h

LA = (2 \(\frac{1}{2}\) cm + 7 \(\frac{1}{2}\) cm + 2 \(\frac{1}{2}\) cm + 7 \(\frac{1}{2}\) cm) ∙ 5 cm

LA = 20 cm ∙ 5 cm

LA = 100 cm^{2}

B = lw

B = 2 \(\frac{1}{2}\) cm ∙ 7 \(\frac{1}{2}\) cm

B = \(\frac{5}{2}\) cm ∙ \(\frac{15}{2}\) cm

B = \(\frac{75}{4}\) cm^{2}

SA = 100 cm^{2} + 2(\(\frac{75}{4}\) cm^{2} )

SA = 100 cm^{2} + 37.5 cm^{2}

SA = 137.5 cm^{2}

The surface area of the right rectangular prism is 137.5 cm^{2}

b. Right triangular prism

Answer:

SA = LA + 2B

LA = P ∙ h

LA = (10 in. + 8 in. + 10 in.) ∙ 12 in.

LA = 28 in. ∙ 12 in.

LA = 336 in^{2}

B = \(\frac{1}{2}\) bh

B = \(\frac{1}{2}\) (8 in.)(9 \(\frac{1}{5}\) in.)

B = 4 in.(9 \(\frac{1}{5}\) in.)

B = (36 + \(\frac{4}{5}\) )in^{2}

B = 36 \(\frac{4}{5}\) in^{2}

SA = 336 in^{2} + 2(36 \(\frac{4}{5}\) in^{2})

SA = 336 in^{2} + (72 + \(\frac{8}{5}\) )in^{2}

SA = 408 in^{2} + 1 \(\frac{3}{5}\) in^{2}

SA = 409 \(\frac{3}{5}\) in^{2}

The surface area of the right triangular prism is 409 3/5 in^{2}2.

Question 2.

Given a cube with edges that are \(\frac{3}{4}\) inch long:

a. Find the surface area of the cube.

Answer:

SA = 6s^{2}2

SA = 6(\(\frac{3}{4}\) in.)^{2}

SA = 6 (\(\frac{3}{4}\) in.) ∙ (\(\frac{3}{4}\) in.)

SA = 6(\(\frac{9}{16}\) in^{2})

SA = \(\frac{27}{8}\) in^{2} or 3 \(\frac{3}{8}\) in^{2}

b. Joshua makes a scale drawing of the cube using a scale factor of 4. Find the surface area of the cube that Joshua drew.

Answer:

\(\frac{3}{4}\) in. ∙ 4 = 3 in.; The edge lengths of Joshua’s drawing would be 3 inches.

SA = 6(3 in.)^{2}

SA = 6(9 in^{2})

SA = 54 in^{2}

c. What is the ratio of the surface area of the scale drawing to the surface area of the actual cube, and how does the value of the ratio compare to the scale factor?

Answer:

54 ÷ 3 \(\frac{3}{8}\)

54 ÷ \(\frac{27}{8}\)

54 ∙ \(\frac{8}{27}\)

2 ∙ 8 = 16. The ratios of the surface area of the scale drawing to the surface area of the actual cube is 16:1. The value of the ratio is 16. The scale factor of the drawing is 4, and the value of the ratio of the surface area of the drawing to the surface area of the actual cube is 4^{2} or 16.

Question 3.

Find the surface area of each of the following right prisms using the formula SA = LA + 2B.

a.

Answer:

SA = LA + 2B

LA = P ∙ h

LA = (12 \(\frac{1}{2}\) mm + 10 mm + 7 \(\frac{1}{2}\) mm) ∙ 15 mm

LA = 30 mm ∙ 15 mm

LA = 450 mm^{2}

B = \(\frac{1}{2}\) bh

B = \(\frac{1}{2}\) ∙ (7 \(\frac{1}{2}\) mm) ∙ (10 mm)

B = \(\frac{1}{2}\) ∙ (70 + 5) mm^{2}

B = \(\frac{1}{2}\) ∙ 75 mm^{2}

B = 75/2 mm^{2}

SA = 450 mm^{2} + 2(\(\frac{75}{2}\) mm^{2})

SA = 450 mm^{2} + 75 mm^{2}

SA = 525 mm^{2}

The surface area of the prism is 525 mm^{2}.

b.

SA = LA + 2B

LA = P ∙ h

LA = (9 \(\frac{3}{25}\) in. + 6 \(\frac{1}{2}\) in. + 4 in.) ∙ 5 in in. ∙ 2 \(\frac{1}{2}\) in.

LA = (\(\frac{228}{25}\) in. + \(\frac{13}{2}\) in. + 4 in.) ∙ 5 in

LA = (\(\frac{456}{50}\) in. + \(\frac{325}{50}\) in. + \(\frac{200}{50}\) in.) ∙ 5 in.

LA = (\(\frac{981}{50}\) in.) ∙ 5 in.

LA = \(\frac{49,050}{50}\) in^{2}

LA = 98 \(\frac{1}{10}\) in^{2}

B = \(\frac{1}{2}\) bh

B = \(\frac{1}{2}\) ∙ 9 \(\frac{3}{25}\)

B = \(\frac{1}{2}\) ∙ \(\frac{228}{25}\) in. ∙ \(\frac{5}{2}\) in.

B = \(\frac{1,140}{100}\) in^{2}

B = 11 \(\frac{2}{5}\) in^{2}

2B = 2 ∙ 11 \(\frac{2}{5}\) in^{2}

2B = 22 \(\frac{4}{5}\) in^{2}

SA = LA + 2B

SA = 98 \(\frac{1}{10}\) in^{2} + 22 \(\frac{4}{5}\) in^{2}

SA = 120 \(\frac{9}{10}\) in^{2}

The surface area of the prism is 120 \(\frac{9}{10}\) in^{2}.

c.

SA = LA + 2B

LA = P ∙ h

LA = (\(\frac{1}{8}\) in. + \(\frac{1}{2}\) in. + \(\frac{1}{8}\) in. + \(\frac{1}{4}\) in. + \(\frac{1}{2}\) in. + \(\frac{1}{4}\) in.) ∙ 2 in.

LA = (1 \(\frac{3}{4}\) in.) ∙ 2 in.

LA = 2 in^{2} + 1 \(\frac{1}{2}\) in^{2}

LA = 3 \(\frac{1}{2}\) in^{2}

B = A_{rectangle} + 2A_{triangle}

B = (\(\frac{1}{2}\) in. ∙ \(\frac{1}{5}\) in.) + 2 ∙ \(\frac{1}{2}\) (\(\frac{1}{8}\) in. ∙ \(\frac{1}{5}\) in.)

B = (\(\frac{1}{10}\) in^{2} ) + (\(\frac{1}{40}\) in^{2} )

B = \(\frac{1}{10}\) in^{2} + \(\frac{1}{40}\)

B = \(\frac{4}{40}\) in^{2} + \(\frac{1}{40}\) in^{2}

B = \(\frac{5}{40}\) in^{2}

B = \(\frac{1}{8}\) in^{2}

SA = 3 \(\frac{1}{2}\) in^{2} + 2(\(\frac{1}{8}\) in^{2}) in^{2}

SA = 3 \(\frac{1}{2}\) in^{2} + \(\frac{1}{4}\) in^{2}

SA = 3 \(\frac{2}{4}\) in^{2} + \(\frac{1}{4}\) in^{2}

SA = 3 \(\frac{3}{4}\) in^{2}

The surface area of the prism is 3 \(\frac{3}{4}\) in^{2}.

d.

SA = LA + 2B

LA = P ∙ h

LA = (13 cm + 13 cm + 8.6 cm + 8.6 cm) ∙ 2 \(\frac{1}{4}\) cm

LA = (26 cm + 17.2 cm) ∙ 2 \(\frac{1}{4}\) cm

LA = (43.2)cm ∙ 2 \(\frac{1}{4}\) cm

LA = (86.4 cm^{2} + 10.8 cm^{2})

LA = 97.2 cm^{2}

SA = LA + 2B

SA = 97.2 cm^{2} + 2(95 cm^{2})

SA = 97.2 cm^{2} + 190 cm^{2}

SA = 287.2 cm^{2}

B = \(\frac{1}{2}\) (10 cm ∙ 7 cm ) + \(\frac{1}{2}\) (12 cm ∙ 10 cm)

B = \(\frac{1}{2}\) (190 cm^{2})

B = \(\frac{1}{2}\) (70 cm^{2} + 120 cm^{2})

B = 95 cm^{2}

The surface area of the prism is 287.2 cm^{2}.

Question 4.

A cube has a volume of 64 m^{3}. What is the cube’s surface area?

Answer:

A cube’s length, width, and height must be equal. 64 = 4 ∙ 4 ∙ 4 = 4^{3}, so the length, width, and height of the cube are all 4 m.

SA = 6s^{2}

SA = 6(4 m)^{2}

SA = 6(16 m^{2})

SA = 96 m^{2}

Question 5.

The height of a right rectangular prism is 4 \(\frac{1}{2}\) ft. The length and width of the prism’s base are 2 ft. and 1 \(\frac{1}{2}\) ft. Use the formula SA = LA + 2B to find the surface area of the right rectangular prism.

Answer:

SA = LA + 2B

LA = P ∙ h

LA = (2 ft. + 2 ft. + 1 \(\frac{1}{2}\) ft. + 1 \(\frac{1}{2}\) ft.) ∙ 4 \(\frac{1}{2}\) ft.

LA = (2 ft + 2 ft. + 3 ft.) ∙ 4 \(\frac{1}{2}\) ft.

LA = 7 ft. ∙ 4 \(\frac{1}{2}\) ft.

LA = 28 ft^{2} + 3 \(\frac{1}{2}\) ft^{2}

LA = 31 \(\frac{1}{2}\) ft^{2}

B = lw

B = 2 ft. ∙ 1 \(\frac{1}{2}\) ft.

B = 3 ft^{2}

SA = LA + 2b

SA = 31 \(\frac{1}{2}\) ft^{2} + 2(3 ft^{2})

SA = 31 \(\frac{1}{2}\) ft^{2} + 6 ft^{2}

SA = 37 \(\frac{1}{2}\) ft^{2}

The surface area of the right rectangular prism is 37 \(\frac{1}{2}\) ft^{2}.

Question 6.

The surface area of a right rectangular prism is 68 \(\frac{2}{3}\) in^{2}. The dimensions of its base are 3 in and 7 in Use the formula SA = LA + 2B and LA = Ph to find the unknown height h of the prism.

Answer:

SA = LA + 2B

SA = P ∙ h + 2B

68 \(\frac{2}{3}\) in^{2} = 20 in. ∙ (h) + 2(21 in^{2})

68 \(\frac{2}{3}\) in^{2} = 20 in. ∙ (h) + 42 in^{2}

68 \(\frac{2}{3}\) in^{2 }– 42 in^{2} = 20 in. ∙ (h) + 42 in^{2 }– 42 in^{2}

26 \(\frac{2}{3}\) in^{2} = 20 in. ∙ (h) + 0 in^{2}

26 \(\frac{2}{3}\) in^{2} ∙ \(\frac{1}{20 \mathrm{in.}}\) = 20 in ∙ \(\frac{1}{20 \mathrm{in.}}\) ∙ (h)

\(\frac{80}{3}\) in^{2} ∙ \(\frac{1}{20 \mathrm{in.}}\) = 1 ∙ h

\(\frac{4}{3}\) in. = h

h = \(\frac{4}{3}\) in. or 1 \(\frac{1}{3}\) in.

The height of the prism is 1 \(\frac{1}{3}\) in.

Question 7.

A given right triangular prism has an equilateral triangular base. The height of that equilateral triangle is approximately 7.1 cm. The distance between the bases is 9 cm. The surface area of the prism is 319 \(\frac{1}{2}\) cm^{2}. Find the approximate lengths of the sides of the base.

Answer:

SA = LA + 2B

LA = P ∙ h

LA = 3(x cm) ∙ 9 cm

LA = 27x cm^{2}

Let x represent the number of centimeters in each side of the equilateral triangle.

B = \(\frac{1}{2}\) lw

B = \(\frac{1}{2}\) ∙ (x cm) ∙ 7.1 cm

B = 3.55x cm^{2}

319 \(\frac{1}{2}\) cm^{2} = LA + 2B

319 \(\frac{1}{2}\) cm^{2} = 27x cm^{2} + 2(3.55x cm^{2})

319 \(\frac{1}{2}\) cm^{2} = 27x cm^{2} + 7.1x cm^{2}

319 \(\frac{1}{2}\) cm^{2} = 34.1x cm^{2}

319 \(\frac{1}{2}\) cm^{2} = 34 \(\frac{1}{10}\) x cm^{2}

\(\frac{639}{2}\) cm^{2} = \(\frac{341}{10}\) x cm^{2}

\(\frac{639}{2}\) cm^{2} ∙ \(\frac{10}{341 \mathrm{~cm}}\) = \(\frac{341}{10}\) x cm^{2} ∙ \(\frac{10}{341 \mathrm{~cm}}\)

\(\frac{3195}{341}\) cm = x

x = \(\frac{3195}{341}\) cm

x ≈ 9.4 cm

The lengths of the sides of the equilateral triangles are approximately 9.4 cm each.

### Eureka Math Grade 7 Module 3 Lesson 21 Exit Ticket Answer Key

Question 1.

Find the surface area of the right trapezoidal prism. Show all necessary work.

Answer:

SA = LA + 2B

LA = P ∙ h

LA = (3 cm + 7 cm + 5 + 11 cm ) ∙ 6 cm

LA = 26 cm ∙ 6 cm

LA = 156 cm^{2}

Each base consists of a 3 cm by 7 cm rectangle and right triangle with a base of 3 cm and a height of 4 cm. Therefore, the area of each base:

B = A_{r} + A_{t}

B = lw + \(\frac{1}{2}\) bh

B = (7 cm ∙ 3 cm) + (\(\frac{1}{2}\) ∙ 3 cm ∙ 4 cm)

B = 21 cm^{2} + 6 cm^{2}

B = 27 cm^{2}

SA = LA + 2B

SA = 156 cm^{2} + 2(27 cm^{2} )

SA = 156 cm^{2} + 54 cm^{2}

SA = 210 cm^{2}

The surface of the right trapezoidal prism is 210 cm^{2}.