A cross-section of a cylinder is nothing but when you cut the slice in a cylinder you can see a circular base in it. That sliced part is known as the cross-section. The top view of the cross-section cylinder looks like a circle. Follow this article to know more about the cross-section and its formulas with examples.
What is Cross Section Cylinder?
A cross-section is a plane section. It means putting an object into pieces creates many parallel cross-sections.
Area of cross section = Ac = π × (D² – d²)/4
Cross Sectional Area of a Cylinder
The cross-sectional area of the cylinder is equal to the area of the circle if it is cut parallel to the circular base. It is obtained by the three-dimensional cylinder. There are two ways to calculate the cross-sectional area of a cylinder they are lateral surface area and total surface area.
Volume of a Cross Section
The volume of the cylinder means calculating the total amount of space occupied by the cross-sectioned part of the cylinder. The volume of the cross-section of the cylinder is the product of the surface area and height of the cylinder.
Formulas on Cross Section
- The volume of a solid figure with uniform cross-section = (Area of the cross-section) × length (or height or breadth)
= A × h - The lateral surface area of a solid figure with uniform cross-section = (Perimeter of the cross-section) × length (or height or breadth)
= P × h - The total surface area of a solid figure with uniform cross-section = Lateral surface area + Sum of the areas of the two plane ends
= P × h + 2 × A
where P = perimeter of the cross-section
h = height
A = Area of the cross-section
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Cross Section Examples
Example 1.
A Hollow pipe of inner radius 3 cm and outer radius 4 cm is melted and changed into a solid right circular cylinder of the same length as that of the pipe. Find the area of the cross-section of the solid cylinder.
Solution:
Given that
Inner radius of the hollow pipe = r = 5 cm
Outer radius of the hollow pipe = R = 6 cm
We know that,
The area of the cross of the pipe = π( R² – r²)
= π(6² – 5²) cm²
= π(36 – 25) cm²
= 9π cm2
Therefore, the volume of the pipe = (Area of the cross section) × length
= 9π × h cm³
(Taking length of the pipe = h cm)
= 9πh cm³
The volume of the right circular cylinder = πr²h cm³
(Taking the radius of the cross section of the cylinder = r cm)
The two volumes are equal so
Therefore, 9πh cm³ = πr²h cm³
r² = 9
Therefore, the area of the cross section of the solid cylinder
=πr² cm²
= 22/7 × 9 cm²
= 28.26 cm²
Example 2.
Find the cross-sectional area of a plane perpendicular to the base of a cube of volume equal to 27 cm³.
Solution:
Given that
Volume of cube = Side³
Therefore,
Side³ = 27
Side = 3 cm
The cross-section of the cube will be a square. Therefore, the side of the square is 3cm.
Hence, cross-sectional area = a² = 3² = 9 sq.cm.
Example 3.
Determine the cross-section area of the given cylinder whose height is 20 cm and radius is 6 cm.
Solution:
Given that
Radius of a cylinder = 6 cm
Height of a cylinder = 20 cm
We know that
When the plane cuts the cylinder parallel to the base, then the cross-section obtained is a circle.
Therefore, The area of a circle, A = πr² square units.
Where
π = 3.14
A = 3.14 (6)² cm²
A = 3.14 (36) cm²
A = 113.04 cm²
Thus, the cross section area of the cylinder is 113.04 cm²
Example 4.
Find the cross-sectional area of a plane perpendicular to the base of a cube of volume equal to 8 cm³.
Solution:
Given that
Volume of cube = Side³
Therefore,
Side³ = 8
Side = 2 cm
The cross-section of the cube will be a square. Therefore, the side of the square is 2 cm.
Example 5.
Determine the cross-section area of the given cylinder whose height is 40 cm and radius is 3 cm.
Solution:
Given that
Radius of a cylinder = 3 cm
Height of a cylinder = 40 cm
We know that
When the plane cuts the cylinder parallel to the base, then the cross-section obtained is a circle.
Therefore, The area of a circle, A = πr² square units.
Where
π = 3.14
A = 3.14 (3)² cm²
A = 3.14 (9) cm²
A = 28.26 cm²
Thus, the cross section area of the cylinder is 28.26 cm²
FAQs on Cross Section of Cylinder
1. What are the units for cross-sectional area?
The units for cross-sectional area is square units.
2. Does a cylinder have a cross section?
The formula at the top of the page can be used to find the volume of a cylinder because a cylinder has a constant cross-sectional area if it is sliced parallel to the circular face.
3. How do you find cross sections?
The volume of any rectangular solid, including a cube, is the area of its base multiplied by its height: V = l × w × h. Therefore, if a cross section is parallel to the top or bottom of the solid, the area of the cross-section is l × w.