A prism is a solid object with identical ends (bases), flat sides and the same cross section all along its length. A prism is named by the shape of its base. Hence, a rectangular prism will have a rectangular base.

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A face is a flat surface of a solid figure. An edge is the place where two faces meet and a vertex (vertices) is a point where two or more edges meet. A rectangular prism has $6$ faces, $12$ edges and $8$ vertices.

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## How to Find

A cross section is the shape made by cutting straight across an object. It is a shape that is formed when you cut through something that is three-dimensional. The cross section of a rectangular prism will be a $2$ dimensional figure.

The cross section of rectangular prism is formed by the intersection of the plane and the prism. When a rectangular prism is cut by a plane parallel to the base of the prism then we will get a rectangle (or square) cross section.

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When a rectangular prism is cut by a plane perpendicular to the base then again we will get a rectangle cross section.

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When a rectangular prism is cut along its diagonal then again the cross section will be a rectangle.

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If a rectangular prism is cut on the top then we will get a triangular cross section as shown in the figure below.

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We can also get a hexagonal cross section by slicing the rectangular prism as follows:

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We should note that since a rectangular prism has six faces, the maximum number of sides of the cross section will be $6$. We cannot get a polygon with seven or more than seven faces by slicing a rectangular prism.

## Examples

Question 1:

What is the area of the different cross sections of the following rectangular prism?

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Solution:

We have $3$ different pairs of bases so we should get $3$ different cross sections in this case.

If we cut the prism by a plane parallel to the base then we will get a rectangle with sides of length $6$ cm and $5$ cm.

So, area of the cross section when cut by a plane parallel to the base will be:

Area $(A)$ = $6 cm \times 5 cm$ = $30 cm^2$

If we cut the prism by a plane perpendicular to the base then we will get a rectangle of side lengths $11$ cm by $5$ cm.

So, area of the cross section when cut by a plane perpendicular to the base will be:

Area $(A)$ = $11 cm \times 5 cm$ = $55 cm^2$
If we cut the prism by a plane across the top then we will get a rectangle of side lengths $11$ cm and $6$ cm.

So, area of the cross section when cut by a plane across the top of the prism will be:

Area $(A)$ = $11 cm \times 6 cm$ = $66 cm^2$
Question 2:

Find the area of the cross section of the following figure?

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Solution:

We can see that the cross section is a rectangle.

If we cut the given prism horizontally then the length and width remains the same.

So, area of the cross section of the given figure will be:

Area $(A)$ = $5 \times 3$ = $15\ units^2$
Question 3:

Describe the shape resulting from the following cross section.

a) image

Solution:

Here, we can see that the rectangular prism is cut horizontally by a plane. Hence, we get the resulting cross sectional shape as a rectangle.

b)   image

Solution:

Here, we can see that the rectangular prism is cut vertically by a plane. Hence, we get the resulting cross sectional shape as a rectangle.