# Volume of prisms

Do you know that transparent glass prisms refract light, and when they do so to white light, they disperse it into various color spectra?

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## What is a prism?

A prism is a 3-dimensional solid which has two opposing surfaces having the same shape and dimension. These opposing surfaces are often referred to as the base and top.

We note that these surfaces may be repositioned such that the top and the base face sideways.

### Types of Prism

There are several types of prisms. Each type is dependent on the shape of the opposing bases. If the opposing bases are rectangular, then it is called a rectangular prism. When these bases are triangular, they are called triangular prisms, and so on.

Below are some types of prisms and their corresponding figures,

• Square prism

• Rectangular prism

• Triangular prism

• Trapezoidal prism

• Hexagonal prism

A diagram showing the types of prisms, StudySmarter Originals

## Volume of prism formula and equation

To find the volume of a prism, you have to take into consideration the base surface area of the prism and the height. Thus, the volume of a prism is the product of its base area and height. So the formula is

$Volum{e}_{prism}=Are{a}_{base}×Heigh{t}_{prism}={A}_{b}×{h}_{p}$

### Application: How to calculate the volume of different types of prisms?

The volume of different types of a prism is calculated using the general rule introduced earlier in the article. Hereafter, we show different direct formulas to compute volumes of different types of prisms.

## Volume of a rectangular prism

A rectangular prism has a rectangular base. It is also called a cuboid.

We recall the area of a rectangle is given by,

$Are{a}_{rec\mathrm{tan}gle}=lengt{h}_{rec\mathrm{tan}gle}×breadt{h}_{rec\mathrm{tan}gle}=l×b$

Thus the volume of a rectangular prism is given by,

$Volum{e}_{rec\mathrm{tan}gularprism}=Are{a}_{base}×Heigh{t}_{prism}=l×b×{h}_{p}$

The length and width of a rectangular matchbox are 12 cm and 8 cm respectively, if its height is 5 cm, find the volume of the matchbox.

Solution:

We first write out the given values,

$l=12cm,b=8cm$ and ${h}_{p}=5cm.$

The volume of the rectangular prism is thus,

${V}_{rec\mathrm{tan}gularprism}=Are{a}_{base}×heigh{t}_{prism}={A}_{rec\mathrm{tan}gle}×heigh{t}_{prism}=l×b×{h}_{p}=12×8×5=480c{m}^{3}.$

## Volume of a prism with triangular base

A triangular prism has its top and base comprising similar triangles.

We recall that the area of a triangle is given by,

$Are{a}_{triangle}=\frac{1}{2}×lengt{h}_{baseoftriangle}×heigh{t}_{triangle}=\frac{1}{2}×{l}_{bt}×{h}_{t}$

Thus, the volume of a triangular prism is given by,

$Volum{e}_{triangularprism}=Are{a}_{traingularbase}×heigh{t}_{prism}=\frac{1}{2}×{l}_{bt}×{h}_{t}×{h}_{p}$

A prism with a triangular base of a length of 10 m and a height of 9 m has a depth of 6 cm. Find the volume of the triangular prism.

Solution:

We first list the given values,

${l}_{bt}=10cm,{h}_{t}=9cm,{h}_{p}=6cm.\phantom{\rule{0ex}{0ex}}$

The volume of the triangular prism is given by

${V}_{prism}=Are{a}_{base}×heigh{t}_{prism}=Are{a}_{triangle}×heigh{t}_{prism}=\frac{1}{2}×{l}_{bt}×{h}_{t}×{h}_{p}=\frac{1}{2}×10×9×6=270c{m}^{3}$.

## Volume of a prism with a square base

All the sides of a square prism are squares. It is also called a cube.

We recall that the area of a square is given by,

$Are{a}_{square}=lengh{t}_{square}×breadt{h}_{square}={\left(lengt{h}_{square}\right)}^{2}$

The volume of a square prism is given by,

$Volum{e}_{squareprism}=Are{a}_{base}×heigh{t}_{prism}=Are{a}_{square}×heigh{t}_{prism}$

But, since this is a square prism, all sides are equal, and hence the height of the prism is equal to the sides of each square in the prism. Therefore,

$heigh{t}_{prism}=lengh{t}_{square}=breadt{h}_{square}$

Thus, the volume of a square prism or a cube is given by,

$Volum{e}_{cube}=Are{a}_{square}×heigh{t}_{prism}=lengt{h}_{square}×heigh{t}_{square}×heigh{t}_{prism}\phantom{\rule{0ex}{0ex}}={l}_{square}×{l}_{square}×{l}_{square}\phantom{\rule{0ex}{0ex}}={\left({l}_{square}\right)}^{3}$

Find the volume of a cube with one of its sides of length 5 cm?

Solution:

We first write out the given values,

${l}_{square}=5cm$

The Volume of a cube is given by,

$Volum{e}_{cube}=Are{a}_{square}×heigh{t}_{prism}=lengt{h}_{square}×heigh{t}_{square}×heigh{t}_{prism}={l}_{square}×{l}_{square}×{l}_{square}$

$={\left({l}_{square}\right)}^{3}={5}^{3}=125c{m}^{3}$

## Volume of a trapezoidal prism

A trapezoidal prism has the same trapezium at the top and base of the solid. The volume of a trapezoidal prism is the product of the area of the trapezium and the height of the prism.

We recall that they are of a trapezium is given by,

$Are{a}_{trapezium}=\frac{1}{2}×heigh{t}_{trapezium}×\left(topbreadt{h}_{trapezium}+downbreadt{h}_{trapezium}\right)\phantom{\rule{0ex}{0ex}}{A}_{trapezium}=\frac{1}{2}×{h}_{t}×\left(t{b}_{trapezium}+d{b}_{trapezium}\right)\phantom{\rule{0ex}{0ex}}$

Thus the volume of a trapezium is given by,

$Volum{e}_{tapezoidalprism}=Are{a}_{trapezium}×heigh{t}_{prism}=\frac{1}{2}×{h}_{t}×\left(t{b}_{trapezium}+d{b}_{trapezium}\right)×{h}_{p}$

A sandwich box is a prism with the base of a trapezium breadths 5 cm and 8 cm with a height of 6 cm. If the depth of the box is 3 cm, find the volume of the sandwich.

Solution:

We first write out the known values, top breadth length is 5 cm, down breadth length is 8 cm, the height of trapezium is 6 cm, and the height of the prism is 3 cm.

Thus, the volume of the trapezoidal prism is given by,

$Volum{e}_{trapezoidalprism}=Are{a}_{trapezium}×heigh{t}_{prism}$

The area of the trapezium can be calculated using the formula,

$A=\frac{1}{2}×{h}_{t}×\left(t{b}_{trapezium}+d{b}_{trapezium}\right)=\frac{1}{2}×6×\left(5+8\right)=3×13=39c{m}^{2}$

Finally, the volume of the trapezoidal prism is

$Volum{e}_{trapezoidalprism}=Are{a}_{trapezium}×heigh{t}_{prism}=39×3=117c{m}^{3}.$

## Volume of a hexagonal prism

A hexagonal prism has both a hexagonal top and base. Its volume is the product of the area of the hexagonal base and the height of the prism.

We recall that the area of a hexagon is given by,

$Are{a}_{hexagon}=\frac{3\sqrt{3}{{l}_{hexagon}}^{2}}{2}$

We note that all sides of a regular polygon are equal. Thus,

$Volum{e}_{hexagonalprism}=Are{a}_{hexagon}×heigh{t}_{prism}=\frac{3\sqrt{3}{{l}_{hexagon}}^{2}}{2}×{h}_{p}$.

A hexagonal prism with one of its sides 7 cm, has a height of 5 cm. Calculate the volume of the prism.

Solution:

We first write out the known values, each side length of the hexagon is 7 cm and the height of the prism is 5 cm.

Thus, the volume of the hexagonal prism is given by,

$Volum{e}_{hexagonalprism}={Area}_{hexagon}×heigt{h}_{prism}$

But,

$Are{a}_{hexagonalbase}=\frac{3\sqrt{3}×{l}^{2}}{2}=\frac{3\sqrt{3}×{7}^{2}}{2}=\frac{3\sqrt{3}×49}{2}=\frac{147\sqrt{3}}{2}c{m}^{2}$

Hence, we have

$Volum{e}_{hexagonalprism}={Area}_{hexagon}×heigh{t}_{prism}=\frac{3\sqrt{3}×{l}^{2}}{2}×{h}_{p}=\frac{147\sqrt{3}}{2}×5=\frac{735\sqrt{3}}{2}c{m}^{3}$

## Examples on volume of prisms

A very useful application of the volume of prisms is the ability to find volumes of different shapes. We will see this in the following example.

Determine the capacity of water that the figure can contain.

Solution:

The figure above consists of two prisms, a rectangular prism at the top and a trapezoidal prism at the base. To find the capacity, we need to find the volume of each.

First, we will calculate the volume of the rectangular prism,

${V}_{rec\mathrm{tan}gularprism}=Are{a}_{rec\mathrm{tan}gle}×heigh{t}_{rec\mathrm{tan}gularprism}=4×5×3=60c{m}^{3}$.

Next, we compute the Volume of the trapezoidal prism,

${V}_{trapezoidalprism}=Are{a}_{trapezium}×heigh{t}_{prism}=\frac{1}{2}×8×\left(5+12\right)×4=\frac{1}{2}×8×17×4=272c{m}^{3}.\phantom{\rule{0ex}{0ex}}$

Then, the volume of the given figure can be calculated,

$Volum{e}_{solid}={V}_{rec\mathrm{tan}gularprism}+{V}_{triangularprism}=60+272=332c{m}^{3}.\phantom{\rule{0ex}{0ex}}$

Therefore, to determine the capacity we need to convert to liters.

Thus,

$1c{m}^{3}=0.001liters\phantom{\rule{0ex}{0ex}}332×0.001=0.332liters.$

## Volume of Prisms - Key takeaways

• A prism is a 3-dimensional solid which has two of its opposing surface the same in both shape and dimension.
• The various types of the prism are based on the shape of the base, such as rectangular, square, triangular, trapezoidal, and polygonal.
• The volume of a regular prism is calculated by finding the product of the base area and the height of the prism.
• Volume of different shapes can be calculated by carrying out simple arithmetic operations on separated regular prisms.

#### Flashcards in Volume of prisms 2

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What is the volume of prism?

The volume of a prism tells us how much it can contain or how much space it will occupy in a 3 dimensional solid.

What is the equation for determining the volume of prism?

The equation for determining the volume of the prism is the Base Area times the Height of the prism.

How do you find the volume of a rectangular prism?

You calculate the volume of a rectangular prism by finding the product of the length, breadth, and height of the prism.

How do you determine the volume of prism with square base ?

You calculate the volume of a prism with a square base by finding the cube of one of its sides.

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