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Cross Section

In geometry, we study about shapes which are usually predefined geometrical figures. Sometimes, one needs to study about the cross section of a shape. The Cross section is one of the important concepts in mathematics. It is defined as the two-dimensional plane obtained by the intersection of a three-dimensional solid shape with the help of a plane. It is quite obvious that cross sectioned surface of a three-dimensional object is a two-dimensional shape.

There are two types of cross sections -
(1) Horizontal Cross Section
(2) Vertical Cross Section

A horizontal cross section is obtained when the plane that passes through the solid object is parallel to its base. On the other hand, a vertical cross section is found when the intersecting plane is perpendicular to the base of the solid. These are known as parallel cross section and perpendicular cross section.

The cross section is the section of the figure after cutting it. For example -
(1)
Any cross section of a sphere is a circle.
(2) A horizontal cross section of a cone is a circle, while its vertical cross section is a triangle.
(3) The horizontal cross section of a cylinder is circle and the vertical cross section is a rectangle.

In this article, we are going to learn about definition of cross sections with suitable problems and figures and their applications. 

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Cross Section Definition

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In geometry the shape obtained by the intersection of a solid by a plane is called as Cross-section. This cross section is mainly happened only on three dimensional geometrical shapes like Rectangular prism, pyramid, triangular prism, pyramid and cube.

For example: Cross section of a sphere is circle.

Definition of Parallel cross section: A plane bisects the solid object in the direction of horizontal that creates the parallel cross section. In the below figure 1 represents the Rectangular prism solid object that can be cut by the plane through the horizontal direction that is parallel. This is outlined as the parallel cross section.

Definition of Perpendicular cross section (vertical cross section): A plane bisects the solid object in the direction of vertical that creates the perpendicular cross section. In the below figure 2 represents the Rectangular prism solid object that can be cut by the plane through the vertical direction that is perpendicular. This is outlined as the perpendicular cross section.

Cross Section Diagram


Cross Section of a Cone

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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 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, an ellipse, a parabola or a hyperbola.
Cone Cross Section

Cross Section Area

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A cross section is the intersection of a figure with a plane. When a plane cuts through an object, an area is projected onto the plane and plane is perpendicular to an axis of symmetry, its projection is called a cross-sectional area.
For example, consider a solid and slice it, then the face you create is known as cross section and the area of the face is known as the cross sectional area.

Cross Section of a Cylinder


For a simple three-dimensional shape, cylinder, the cross-sectional projection is a circle when it is sliced parallel to its base, and the area is easy to calculate. The formula to calculate this cross sectional area is $\pi$ multiplied by the square of radius of the cylinder.

Volume by Cross Section

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The volume of a solid is the integral of the area of the cross section. Let V be the volume of a solid bounded by planes that are perpendicular to the x-axis at x = a and x = b. If A(x) is the area of a cross section of the solid formed by a plane perpendicular to the x-axis at any point and A(x) is continuous in [a, b], then the volume of the solid is

V = $\int_a^b$ A(x) dx

Let us find the volume of the solid with square cross section perpendicular to the x-axis and of side length $x^2$ for 0 $\leq$ x $\leq$ 2.

The cross section at x has area $(x^2)^2$, so the volume is

V = $\int_0^2$ $(x^2)^2$ dx = $\int_0^2$ $x^4$ dx

= $\frac{x^5}{5}$$|_0^2$

= $\frac{32}{5}$

The volume of a solid is $\frac{32}{5}$ cubic units.

Circular Cross Section

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A cross section is the intersection between a solid object and a cutting plane. A plane is infinitely thin, the resulting intersection of the two planes is a planar section. Most of the objects have several cross-sections associated with them. Below we can see few of the circular cross sections of the different solids. image

Cross Sections Example Problems

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Below are the examples on Cross Section -

Problem 1:

See the below diagram and then find the type of the cross section and identify the part of the parallel cross section

Solution: Here the given diagram looks like as the pentagonal prism. This solid is uniformly cut by the plane and it’s parallel to the base. So it is comes under the type of the parallel cross section. This cross section is same to the figure 2. Because the original figure of the parallel cross section of pentagonal prism is the pentagon.

Problem 2:

Which is the cross section of the given rectangular prism, after cutting vertically?

Option:

Cross Section

Solution:

A is correct (because cross section of the given figure is a rectangle).

Problem 3:

Find the area of cross section of the given figure. 

Cross Section Geometry

Solution:

From the given figure: Cross section of the figure is rectangular.
If we cutting any figure horizontally then their is no change in length and width.

So the area of the cross section = Length * Width

= 5m * 3m

= 15 m2

 Therefore, the area of cross section is 15 m2.

Problem 4:

Find the area of cross section of the given figure. Where radius = 5 cm and height = 30 cm?

Cross Section example

Solution:

The cross section area of a cylinder after cutting it vertically is a circle.

The area of a circle = $\pi$r2

So area of cross section is = 3.14 * 52 cm2

= 3.14 * 25 cm2

= 75.5 cm2

Thus the cross section area of the cylinder is 75.5 cm2.

More topics in Cross Section
Cross Section of a Cone Cross Section Word Problems
NCERT Solutions
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