Volume of Pyramid

Volume of Pyramid

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Introduction to Pyramids

Have you visited or seen pictures of the Great Pyramid of Giza

What shape is it?

Shape of a pyramid-real life example

Pyramids are solid shapes.

They have a polygon as their base and triangular faces that meet at the apex.

Apex is a vertex which is opposite to the base of the pyramid.

A square pyramid with apex and base


Types of Pyramids

We just learnt that a pyramid has a polygon as its base.

Pyramids can be classified into different types depending on their bases. They include:

  1. Triangular Pyramid
  2. Quadrilateral Pyramid
  3. Pentagonal Pyramid

Let's discuss explore them further.

Triangular Pyramid

When the base of a pyramid is a triangle, we call it a triangular pyramid.

A triangular pyramid

Note: Tetrahedron is a triangular pyramid which has all its faces as triangles.

Quadrilateral Pyramid

A Quadrilateral pyramid has quadrilaterals like a square or a rectangle as its base.

A rectangular pyramid

A quadrilateral pyramid with a square base

Pentagonal Pyramid

A pentagonal pyramid has a pentagon as its base.

A pentagonal pyramid

If we know the area of a pentagon, we can calculate the volume of a pentagonal pyramid.

Here's a task for you. Can you draw a hexagonal pyramid?


Definition of Volume of a Pyramid

Volume of a pyramid refers to the space enclosed between its faces.

Look at the simulation below where the nets of different pyramids fold to form a solid shape.

The amount of space enclosed by these faces is the volume of the pyramid.

The volume of a pyramid is one-third of the product of the area of the base and the height of the pyramid.

\(\begin{align}\text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height}\end{align}\)

Consider a square pyramid.

We know that the base of this pyramid is a square.

Let the side of the square be \(a\) and height of the pyramid be \(h\).

Thus, the volume of the square pyramid is:

\(\begin{align}\text{Volume} = \frac{1}{3} \times a^2 \times h \end{align}\)

We can use the same method to find the volume of a pyramid with any polygon as its base.

We know that the area of a polygon is given by the formula:

  \(\begin{align}\frac{1}{2} \times \text{Perimeter} \times \text{apothem} \end{align} \)

Note that apothem is the line segment from the center of a regular polygon to the midpoint of a side.

Once we find the base area of the polygon, we can apply the volume of a pyramid formula to calculate it.

Can you deduce a formula to calculate the volume of a pentagonal pyramid and a hexagonal pyramid?

Note that area of a pentagon is

\(\begin{align}\frac{1}{2} \times 5 \times (\text{edge length}) \times \text{apothem }\end{align}\)

(\(\because\! \text{Perimeter of pentagon} \!=\!5\!\!\times\!\!\text{edge length} \))

Therefore, Volume of a Pentagonal pyramid is

\(\begin{align}\text{Volume} = \frac{5}{6} \times a \times s \times h \end{align} \)

Area of a hexagon is

\(\begin{align}\frac{1}{2} \times 6 \times (\text{edge length }) \times  \text{apothem }\end{align} \)

(\(\because \text{Perimeter of hexagon} \!=\!6 \!\times\! \text{edge length} \))

Therefore, Volume of a Hexagonal pyramid is

\(\begin{align}\text{Volume} = a \times s \times h \end{align} \)

Where;

  • \(a\) = apothem length
  • \(s\) = edge length of the polygon and
  • \(h\) = height of the pyramid

If apothem length is not known,

We can also use the area of a triangle formula to calculate the area of a polygon.

Note that a regular polygon is divided into \(n\) equilateral triangles where \(n\) is the number of sides in a polygon.

Pyramids can also be irregular.

If the base polygon is irregular, the pyramid is called an irregular pyramid.

By default, a pyramid is assumed to be a regular pyramid unless explicitly mentioned as irregular.

You can download the volume of a pyramid worksheet at the end of this page.

Improve your knowledge on the volume of a pyramid with examples from our experts.


Volume of a Pyramid Calculator

Use the calculator below to find the volume of a pyramid.

Select the type of Pyramid, enter the values and click Go.

 

 
Thinking out of the box
Think Tank
  1. Can you deduce a formula to find the volume of a pyramid whose base is a decagon?
  2. What is the difference between a regular triangular pyramid and a regular tetrahedron?
  3. Is cone a pyramid?

Help your child score higher with Cuemath’s proprietary FREE Diagnostic Test. Get access to detailed reports, customised learning plans and a FREE counselling session. Attempt the test now.

Solved Examples

Example 1

 

 

Find the volume of a regular square pyramid with base sides 5 cm and height 9 cm.

A quadrilateral pyramid with a square base

Solution:

Pyramid volume  = \(\begin{align}\frac{1}{3} \times a^2 \times h \end{align}\) where \(a\) is the length of the edge of the square.

Given, 

  • a = 5 cm
  • h = 9 cm

Substituting the values, we get:

\[\begin{align} \text{Volume} &= \frac{1}{3}  \times 5^2 \times 9 \\
&= \frac{1}{3} \times 25 \times 9 \\
&=75\: \text{cm}^3 \end{align} \]

\(\therefore\) Volume of the pyramid = 75 cm3
Example 2

 

 

Cheops Pyramid in Egypt has a base measuring about 755 ft. \(\times\) 755 ft. and the height is around 480 ft.

Calculate its volume.

 Volume of Cheops Pyramid which is a square pyramid

Solution:

Since the base measures 755 ft. \(\times\) 755 ft., the Cheops Pyramid is a square Pyramid.

Volume of a square pyramid = \(\begin{align}  \frac{1}{3} \times a^2 \times h \end{align}\)

Substituting the values, we get:

\[\begin{align} \text{ Pyramid Volume} &=\frac{1}{3} \times 755 \times 755   \times 480 \\
&= \frac{1}{3} \times 570025 \times 480 \\
&=91204000 \:\text{cubic feet} \end{align} \]

\(\therefore\) Volume of Cheops Pyramid = 9,12,04,000 cubic ft.
Example 3

 

 

A pyramid has a hexagon as its base of edge length 6 cm and height 9 cm.

Find its volume.

Find the volume of a hexagonal pyramid

Solution:

Volume of a hexagonal pyramid

\(\begin{align}\frac{\sqrt{3}}{2} \times a^2 \times h \end{align}\)

Given, 

  • a = 6
  • h = 9

Substituting the values, we get:

\[\begin{align} &= \frac{\sqrt{3}}{2}  \times 6 \times 6   \times 9 \\
&= \frac{\sqrt{3}}{2} \times1944  \\
&\approx 280.59\: \text{cm}^3 \end{align} \]

\(\therefore\)Volume of hexagonal pyramid = 280.59 cm3
Example 4

 

 

Which has a greater volume: A square pyramid of base 12 cm and height 10 cm or A pentagonal pyramid of base area 144 cm2 and height 10 cm?

Compare the volume of a quadrilateral pyramid with a square base with that of a pentagonal pyramid

Solution:

Volume of the Square Pyramid

\(\begin{align}  \frac{1}{3} \times a^2 \times h \end{align}\) 

where \(a\) is the length of the edge of the square.

Given, 

  • a = 12 cm
  • h = 10 cm

Substituting the value, we get:

Volume of the square pyramid:

\[\begin{align} &= \frac{1}{3}  \times 12^2 \times 10 \\
&= \frac{1}{3} \times 144 \times 10 \\
&=480 \: \text{cm}^3 \end{align} \]

Volume of a Pentagonal Pyramid:

\(\begin{align} \frac{1}{3} \times \text{Base Area} \times \text{Height}\end{align}\)

Given,

  • base area = 144 cm2
  • height = 10 cm

Substituting the values,

Volume of a pentagonal pyramid:

\[\begin{align} &=\frac{1}{3}  \times 144 \times 10 \\
 &=480 \: \text{cm}^3 \end{align} \]

\(\therefore\) Volume of both pyramids are equal
Example 5

 

 

Tim built a rectangular tent for his night camp.

The base of the tent is a rectangle of side 6 m \(\times\) 10 m and the height is 3 m. What is the volume?

A tent in the shape of a rectangular pyramid measuring 6 m by 10 m by 3 m

Solution:

Volume of a rectangular pyramid is

\(\begin{align}\frac{1}{3} \times l \times b \times h \end{align}\)

where \(l\) and \(b\) are the length and breadth of the rectangle.

Given,

  • \(l\) = 6 m
  • \(h\) = 3 m 
  • \(b\) = 10 m

Substituting the values, we get, pyramid volume:

\[\begin{align} &=\frac{1}{3}  \times 6 \times 10  \times 3 \\
&= \frac{1}{3} \times 180 \\
&=60 \: \text{m}^3 \end{align} \]

\(\therefore\) Volume of the tent is 60 m3

You can use the volume of a pyramid worksheet at the end of this page to solve more problems.
 

Have a doubt that you want to clear? Get it clarified with simple solutions on Volume of a Pyramid from our Math Experts at Cuemath’s LIVE, Personalised and Interactive Online Classes.

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Practice Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 

 
 
 
 
 
 
 
important notes to remember
Important Notes

Volume of a pyramid formula with height \(h\) units and when the base is a:

  • Square : \(\frac{1}{3} \times s^2 \times h \) where \(s\) is edge length of the square.
  • Rectangle : \( \frac{1}{3} \times lbh \) where \(l\) and \(b\) are the length and breadth of the rectangle.
  • Pentagon : \(\frac{5}{6} \times ash \) where \(a\) is the apothem length and \(s\) is the edge length.
  • Hexagon  :  \( \frac{\sqrt{3}}{2} \times s^2 \times h \)  or \(a.s.h\) where \(s\) is the edge length and \(a\) is the apothem.  

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions(FAQs)

1. How do you find the volume of a pyramid?

Volume of a pyramid is one third the base area multiplied to the height of the pyramid.

Vpyramid = \(\begin{align}\frac{1}{3} \times \text{Base area} \times \text{Height}\end{align}\)

2. Is the volume of a pyramid a cube?

Volume is measured in cubic units.

For example, the volume of a square pyramid of base area 90 cmand height 10 cm is 300 cubic centimeter or 300 cm3

3. How do you calculate the volume of a triangular pyramid?

Volume of a triangular pyramid is one third the base area multiplied to the pyramid's height. 

Vpyramid =  \(\begin{align}\frac{1}{3} \times \text{Base area} \times \text{Height}\end{align}\)

  
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