C**ross section **means the representation of the intersection of an object by a plane along its axis. A cross-section is a shape that is yielded from a solid, when intersected by a plane. For example, a cylinder-shaped object is cut by a plane parallel to its base; then the resultant cross-section will be a circle.

Also, you will see some real-life examples of cross-sections such as a tree after it has been cut, which shows a ring shape. If cut a cubical box by a plane parallel to its base, then we obtain a square.

## 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 sectional area of different solids is given here with examples. Let us figure out the cross-sections of cube, sphere, cone and cylinder here.

### Cross-Sectional Area

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-sectional area.

**Example: Find the cross-sectional area of a plane perpendicular to the base of a cube of volume equal to 27 cm****3****.**

Solution: Since we know,

Volume of cube = Side3

Therefore,

Side3 = 27 [Given]

Side = 3 cm

Since, 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.

**Volume by Cross Section**

Since the cross section of a solid is a two-dimensional shape, therefore, we cannot determine its volume.

## 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.

### Solved Problem

**Problem: **

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

**Solution:**

Given:

Radius = 4 cm

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 = πr^{2} square units.

Take π = 3.14

Substitute the values,

A = 3.14 (4)^{2 } cm^{2}

A = 3.14 (16) cm^{2}

A = 50.24 cm^{2}

Thus, the cross section area of the cylinder is 50.24 cm^{2}