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Volume of a Truncated Pyramid

In geometry, we come across with the truncated solid figures at times. They are also termed as the frustums. By a frustum, we mean to refer to a solid shape which is being cut from the top via a plane that is parallel to the base.

The most common frustums used in geometry are pyramid frustums and cone frustums. 
The frustums are seen in computer graphics in order to define a 3D region that is visible on the screen. They are commonly used in aerospace industry in fairing of a multistage rocket between two stages.

A truncated pyramid is a kind of frustum that is obtained from a pyramid. In this article, we are going to discuss a truncated pyramid. Basically, we shall focus on the volume of truncated pyramid and method of finding it. Also, we will understand this concept with the help of solved examples.

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Definition

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Suppose you have a pyramid-shaped object. Take a knife and cut down some portion of its vertex. In this way, the shape you will obtain is called a truncated pyramid. Refer the following image, the lower portion will be the truncated pyramid.

Volume of a Truncated Pyramid

Mathematically, A pyramid frustum or a truncated pyramid is defined as a kind of shape which is obtained when the vertex of a pyramid is cut away using a plane that goes parallel to the base of pyramid. We get a shape similar to the following image depending upon the type of base.

Pyramid Frustum

Formula

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How to find the volume of a truncated pyramid ? For this, you need to know the volume of the whole pyramid. The volume of a pyramid can be calculated using the following formula :

Volume = $\frac{1}{3}$ area of base $\times$ height of pyramid

The volume of truncated pyramid is determined by subtracting the volume of cut out top from the volume of the whole pyramid. Suppose that the height and area of base of the whole pyramid be $h_{1}$ and $A_{1}$, while those of cut-off pyramid be $h_{2}$ and $A_{2}$, then we may write the formula for volume of truncated pyramid in the following way :

Volume of truncated pyramid = $\frac{1}{3}$ $A_{1}\ h_{1}$ - $\frac{1}{3}$ $A_{2}\ h_{2}$

Volume of truncated pyramid = $\frac{1}{3}$ $(A_{1}\ h_{1}\ -\ A_{2}\ h_{2})$


Let us consider a special case of square pyramid which has square base. Let us take a look at the following diagram.

Volume of a Truncated Pyramid Formula

The height of the truncated pyramid is h. The side of larger base be a and smaller base be b. Then the formula for its volume is given below.

Volume of truncated square pyramid = $\frac{1}{3}$$(a^{2}\ +\ ab\ +\ b^{2})\ h$

How to Calculate Volume of Truncated Pyramid

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To calculate the volume of truncated pyramid, the following method is used.

(1) Determine the heights of the whole pyramid and cut-off pyramid. Denote them by $h_{1}$ and $h_{2}$.

(2)
 Calculate the areas of both bases and denote them by $A_{1}$ and $A_{2}$.The formulae for areas of bases will depend on upon the kind of base, i.e. rectangular, pentagonal etc.

(3) Substitute all the values in the formula$V$ = $\frac{1}{3}$ $(A_{1}\ h_{1}\ -\ A_{2}\ h_{2})$ and calculate the volume.

Truncated Square Pyramid

(1) Determine sides of both square bases and denote them by a and b.

(2) Find the height h of the truncated pyramid.

(3) Plug these values in the formula $V$ = $\frac{1}{3}$$(a^{2}\ +\ ab\ +\ b^{2})\ h$ and calculate the required value of volume.

Examples

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Consider the following examples.

Example 1 : Find the volume of a truncated pyramid whose base areas are 45 cm$^{2}$ and 36 cm^{2}. The height of the pyramid from which it is cut was 21 cm and the height of cut-off portion was 11 cm.

Solution :
 Given that $A_{1}$ = 45 cm$^{2}$ and $A_{2}$ = 36 cm$^{2}$.

$h_{1}$ = 21 cm, $h_{2}$ = 11 cm

The formula for volume of truncated pyramid is

$V$ = $\frac{1}{3}$ $(A_{1}\ h_{1}\ -\ A_{2}\ h_{2})$

$V$ = $\frac{1}{3}$ $(45\ \times\ 21\ -\ 36\ \times\ 11)$

$V$ = $\frac{1}{3}$ $(945\ -\ 396)$

$V$ = $\frac{1}{3}$ $\times\ 549$

$V$ = $183$ $cm^{2}$
Example 2 : The sides of bases of a truncated square pyramid measure 4 inches and 7 inches. If the height of this pyramid is 10 inches, calculate its volume.

Solution :
 Given that

$a$ = 7 in, $b$ = 4 in, $h$ = 10 in

Substituting above values if the formula of volume of truncated square pyramid is given by :

$V$ = $\frac{1}{3}$$(a^{2}\ +\ ab\ +\ b^{2})\ h$

$V$ = $\frac{1}{3}$$(7^{2}\ +\ 7\ \times\ 4\ +\ 4^{2})\ \times\ 10$

$V$ = $\frac{1}{3}$$(49\ +\ 28\ +\ 16)\ \times\ 10$

$V$ = $\frac{1}{3}$$(93)\ \times\ 10$

$V$ = $310\ in^{2}$

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