Cross Section

Cross Section Meaning

A cross-section is the intersection of a given object, by a plane along its axis. In geometry, a cross-section is a non-empty intersection of a solid body by a plane. Snipping an object into slices creates many parallel cross-sections which further creates multiple shapes, like a circle, rectangle, etc.

It is not necessary for the object to be three dimensional, this concept can be applied for two-dimensional shapes as well.

For example, if a plane cuts through mountains on a two-dimensional map, the result is showing points on the surface of the mountains of equal elevation.

Also, you can understand the concept of the cross-section by learning it through real-time examples, such as a tree after it has been cut shows a ring shape, this helps us in understanding a smaller portion by magnifying it. 

In geometry, we learn different types of shapes, single-dimensional shape, two-dimensional shape, and so on. The concept of cross-sections plays an important role in understanding these shapes better, hence we would move ahead in understanding cross-sections conceptually by its definition, types, and relative examples.

What is Cross Section?

A cross-section is defined as the shape obtained by the intersection of a solid object by a plane. The cross-section of a three-dimensional shape is a two-dimensional geometric shape.

Types of Cross Sections

There are two major types of cross-section, namely

* Horizontal cross-section

* Vertical cross-section

Horizontal Cross Section

The horizontal cross-section also known as parallel cross-section is when a plane cuts a solid shape in the horizontal direction such that it creates a parallel cross-section with the base. For example, the horizontal cross-section of a cylinder is a circle.

Vertical Cross Section

In perpendicular cross-section, a plane cuts the solid shape in the vertical direction which is perpendicular to its base. For example, the vertical cross-section of a cylinder is a rectangle. 

Cross Section Examples

Some widely understood examples of cross-sections are

* Any cross-section of the sphere may be a circle

* The vertical cross-section of a cone may be a triangle, and therefore the horizontal cross-section may be a circle

* The vertical cross-section of a cylinder may be a rectangle, and therefore the horizontal cross-section may be a circle

Cross Section in Geometry

Before we understand the different cross-sections of a three-dimensional shape, we would understand the important terminologies, and functions relative to it.

Cross Section Area and Volume

What is Cross Sectional Area?

When a solid object is intersected by a plane, an area is projected onto the plane. This plane is further perpendicular to the axis of its symmetry; thus, this projection is called a cross-sectional area.

The cross section area formula = l x w

The volume of Cross Section

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

Cross Section of a Cone

A cone is a pyramid-like structure with a circular base and a triangular top. The cross-section of a cone is called a conic section, this is achieved by depending on the relationship between the plan and slant surface. It could be a circle, a parabola, an ellipse.

Cross Sections of a Cylinder

A cylinder could produce different shapes depending on the way its cut, it could produce a circle, rectangle, or oval.

If the cylinder has a horizontal cross-section, then the shape would be a circle. If the plane of the base of the cylinder is perpendicular to its base, then the shape would be a rectangle.

The oval shape is obtained, when the plane intersects the cylinder parallel to the base with variation in its angle.

Cross Section of Sphere

A sphere is a round solid figure, whose surface, with every point on its surface equidistant from its centre.

A sphere has the littlest area for its volume. The intersection of a plane in a sphere produces a circle, likewise, all cross-sections of a sphere are circles.

Cross Section Example Solved Problem

Problem 1: Determine the cross-section area of the given cylinder whose height is 25 cm and the radius is 4 cm.

Solution:

Given:

Radius = 4 cm

Height = 25 cm

We know that when the plane cuts the cylinder parallel to the bottom, then the cross-section obtained may be 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

The area of the cylinder is = 50.24 cm

Problem 2: A solid, metallic, right circular cylindrical block of radius 7 cm and height 8 cm is melted, and small cubes of edge 2 cm are made from it. How many such cubes can be made from the block?

Solution:

For the proper circular cylinder, we've radius (r) = 7 cm, height (h) = 8 cm.

Therefore, its volume = πr2h

 = 3.14 × 72 × 8 cm3

 = 1232 cm3