 # Cross Section

Cross section means the representation of the intersection of an object by a plane along its axis. For example, a cylinder-shaped object is cut by a plane parallel to its base; then the resultant cross-section will be a circle. So, there has been an intersection of the object. It is not necessary that the object has to be three-dimensional shape; instead, this concept is also applied for two-dimensional shapes.

Also, you will see some real-life examples of cross-sections such as a tree after it has been cut, shows a ring shape. So, we can say this method helps to see a small portion of anything which is big.

In Geometry, we study different types of geometric shapes, such as two-dimensional shapes and three-dimensional shapes. The cross-section is one of the important concepts in Geometry. In this article, let us discuss in detail about the definition, types and cross-section of different shapes along with examples.

## Cross Section Definition

In Geometry, the cross-section is defined as the shape obtained by the intersection of solid by a plane. The cross-section of three-dimensional shape is a two-dimensional geometric shape. In other words, the shape obtained by cutting a solid parallel to the base is known as a cross-section.

### Types of Cross Section

The cross-section is of two types, namely

• Horizontal cross-section
• Vertical cross-section

Horizontal or Parallel Cross Section

In parallel cross-section, a plane cuts the solid shape in the horizontal direction (i.e., parallel to the base) such that it creates the parallel cross-section

Vertical or Perpendicular Cross Section

In perpendicular cross-section, a plane cuts the solid shape in the vertical direction (i.e., perpendicular to the base) such that it creates a perpendicular cross-section

### Cross-section Examples

The examples for cross-section for some shapes are:

• Any cross-section of the sphere is a circle
• The vertical cross-section of a cone is a triangle, and the horizontal cross-section is a circle
• The vertical cross-section of a cylinder is a rectangle, and the horizontal cross-section is a circle

## Cross-sections in Geometry

The cross-sections for different 3D shapes are given here. Before that, we are going to learn two important terminologies such as

### Cross Section Area and Volume

When a plane cuts a solid object, an area is projected onto the plane. That plane is then perpendicular to the axis of symmetry. Its projection is known as the cross-section area.

Volume by Cross Section

The volume of the solid is defined as the integral of the area of the cross-section.

### Cross Sections of Cone

A cone is considered a pyramid with a circular cross-section. Depending upon the relationship between the plane and the slant surface, the cross-section or also called conic sections (for a cone) might be a circle, a parabola, an ellipse or a hyperbola. Also, see: Conic Sections Class 11

### Cross Sections of cylinder

Depending on how it has been cut, the cross-section of a cylinder may be either circle, rectangle, or oval. If the cylinder has a horizontal cross-section, then the shape obtained is a circle. If the plane cuts the cylinder perpendicular to the base, then the shape obtained is a rectangle. The oval shape is obtained when the plane cuts the cylinder parallel to the base with slight variation in its angle ### Cross Sections of Sphere

We know that of all the shapes, a sphere has the smallest surface area for its volume. The intersection of a plane figure with a sphere is a circle. All cross-sections of a sphere are circles. ### Cross Sections Solved Problem

Problem:

Determine the cross-section area of the given cylinder whose height is 25 cm and radius is 4 cm.

Solution:

Given:

Height = 25 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.

Take π = 3.14

Substitute the values,

A = 3.14 (4)2 cm2

A = 3.14 (16) cm2

A = 50.24 cm2

Thus, the cross section area of the cylinder is 50.24 cm2