Cross Section

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 for Different Shapes

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:

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 of the conic section may be a circle, a parabola, an ellipse or a hyperbola.

Cross Section of Cone

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 Section of Cylinder

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 Section of Sphere

Cross Sections Example 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 = 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