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
FAQs on Cross Section
1. Whats is cross-section? Give an example.
A cross-section is the intersection of a given object, by a plane along its axis. In geometry, a cross-section in 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.
