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.