Cross section is the intersection of a three dimensional body with a plane. It is the shape made when a solid is cut through parallel to the x-axis or to the y-axis. It can be any view of something which shows a representative portion of each of its parts. In solids, cross sections of equal areas will have equal volume by cavalien's principle.

## Cross Section Diagram

A cross section diagram is what you could see when there is a cut through the object and see what the new surface looks like. Cross section views have diagonal line showing the new cutting plane.
Given below are the images of cross section:

## Cross Section Area

Cross sectional area is the area of the resulting face of a three dimensional object, when an object is sliced perpendicular to the specified axis at a specified point. For example, for a cylinder, cross sectional projection will be a circle, when it is sliced parallel to its base. The area of the face(cross section) is known as the cross sectional area.

## Cross Section of a Cube

If you cut the cube parallel and perpendicular to the faces, the cross sectional area will be a square, having the length same as the length of the cube. Depending on where the cut is made on the cube, there is a possibility of forming a rectangle, triangle or a hexagon etc.

## Cross Section of a Cone

The cross section is the shape obtained by a cut through a solid perpendicular to the length. A cone is pyramid with a circular cross section. The double cone is a quadratic surface and each single cone is called a nappe.

The cross section of a conic section depending upon the relationship between the plane and the slant surface of the cone, the section may be a circle, a parabola, an ellipse or a hyperbola.

## Cross Section of a Pyramid

A pyramid is a polyhedron formed by connecting a polygonal base and a point called apex. It is a conic solid with polygonal base.

Given below are the three cross sections of a pyramid with a square base.
 If the pyramid is cut with a plane passing through the top vertex and perpendicular to the base, the intersection of the pyramid and plane results in a triangular cross section. If the pyramid is cut with a plane perpendicular to the base, the intersection of the pyramid and the plane results in a trapezoidal cross section When a pyramid is cut on a plane parallel to the base, the intersection of the pyramid and the plane results in a square cross section.

## Volume by Cross Section

Volume is the quantity of a three dimensional space enclosed in a close boundary. For example, volume of a container is understood as the capacity of the container. Volume of solids with known cross sections on an interval can be found by using definite integral.

The formula to find the volume of the solid on the interval [a, b] is V = $\int_{a}^{b}$ A(x) dx
Here, the cross sections are perpendicular to the x-axis. So, their areas are the functions of x denoted by A(x).

If the cross section is perpendicular to the y-axis, areas will be functions of y, denoted by A(y) and the volume of the solid on [a, b] is V = $\int_{a}^{b}$ A(y) dy

## Cross Sections Example Problems

Given below are some of the examples on cross sections.

### Solved Examples

Question 1: Find the area of cross section, when a cylinder is cut vertically parallel to the base, whose radius is 3 cm and height is 25 cm.
Solution:
Given: The cylinder is cut vertically. This means the obtained figure will be a circle.
Area of a circle is $\pi$ $\times$ r$^{2}$

So, the area of cross section = 3.14 $\times$ 3$^{2}$ = 28.26

Therefore, cross section of the cylinder = 28.26 cm$^{2}$

Question 2: Find the volume of the solid, whose base is bounded by the circle x$^{2}$ + y$^{2}$ = 25$^{2}$ and the cross sections perpendicular to the x-axis are squares.
Solution:
We know that, area of square, A = y$^{2}$
where, y is the length of the side

Given: x$^{2}$ + y$^{2}$ = 25$^{2}$

The formula to find the volume of the solid on the interval [a, b] is

V = $\int_{a}^{b}$ A(x) dx

V = $\int_{-5}^{5}$ y$^{2}$ dx

V = 2 $\int_{0}^{5}(625 - x^{2})$ dx

V = 2 [(625x - $\frac{x^{3}}{3}$)]$_{0}^{5}$

V = 2[3125 - 41.67]
V = 6166.66
Therefore, the volume of the solid is 6166.66 cubic units.