Volume of a Pyramid(Formula, Derivation & Examples)

A pyramid is a 3D shape whose base is a polygon and whose side faces (that are triangles) meet at a point which is called the apex (or) vertex. The space occupied by a pyramid in a three-dimensional space is called the volume of a pyramid and it depends on its base area and height. Volume of a pyramid is measured in terms of cubic units, such as $cm^{3}$, $m^{3}$, $mm^{3}$, $in^{3}$, $ft^{3}$, etc.

Let’s learn how to find the volume of a pyramid and its formula.

What is the Volume of a Pyramid?

The volume of a pyramid is the space enclosed between its triangular faces. It is measured in cubic units such as $cm^{3}$, $m^{3}$, $in^{3}$, etc. 

A pyramid is a three-dimensional shape where its base (a polygon) is joined to the vertex (apex) with the help of triangular faces. The perpendicular distance from the apex to the center of the polygon base is referred to as the height of the pyramid. 

A pyramid’s name is derived from its base. For example, a pyramid with a square base is referred to as a square pyramid. The volume of the pyramid is calculated by finding one-third of the product of the base area times its height.

$\text{Volume of a Pyramid} = \frac {1}{3} \times \text{Area of Base} \times \text{Height of Pyramid}$.

The general formula for finding the area of a $n$-sided polygon is given by $\text{Area} = \frac {ns^{2}}{4 \tan \frac {180}{n}}$

where $n$ is the number of edges (or sides) in a polygon

$s$ is the length of the edge(or side) of a polygon

Therefore, the formula for volume of a $n$-sided pyramid of edge length $s$ and height $h$ is given by $\text{Volume} = \frac {1}{3} \times \frac {ns^{2}h}{4 \tan \frac {180}{n}} = \frac {ns^{2}h}{12 \tan \frac {180}{n}}$.

Volume of Pyramid Formula

These are the formulas for different pyramids.

Pyramid TypeNumber of sides in Base $n$Volume Formula
Triangular Pyramid$3$$\frac {s^{2}h}{4 \tan 60^{\circ}}$
Square Pyramid$4$$\frac {s^{2}h}{3\tan 45^{\circ}}$
Pentagonal Pyramid$5$$\frac {5s^{2}h}{12 \tan 36^{\circ}}$
Hexagonal Pyramid$6$$\frac {s^{2}h}{2 \tan 30^{\circ}}$
Heptagonal Pyramid$7$$\frac {7s^{2}h}{12 \tan 25.71^{\circ}}$
Octagonal Pyramid$8$$\frac {2s^{2}h}{3 \tan 22.5^{\circ}}$
Nonagonal Pyramid$9$$\frac {3ns^{2}h}{4 \tan 20^{\circ}}$
Decagonal Pyramid$10$$\frac {5s^{2}h}{6 \tan 18^{\circ}}$

Note: 

  • The above formulae are applicable only when bases are regular polygons.
  • In case the bases are irregular polygons, then first find one-third of the area of the base and multiply it by the height of the pyramid to find the volume.

Examples

Ex 1: What is the volume of a pyramid with a height of $15 m$ and a square base with a side length of $7 m$?

The side length of the base = $7 m$

Area of base = $\text{B} = 7^{2} = 49 m^{2}$.

The height of a pyramid $\text{H} = 15 m$.

Volume of a pyramid = $\frac {1}{3} \text{BH} = \frac {1}{3} \times 49 \times 15 = 245 m^{3}$.

Ex 2: Find the volume of the following pyramid.

volume of a pyramid

It’s a square pyramid.

Length of a side of a base = $10 m$.

Area of the base $\text{B} = 10^{2} = 100 m^{2}$.

volume of a pyramid

The height of the pyramid is $AM = H$.

To find the height $H$, first, find the length of the diagonal $CE$ of the square base.

Using Pythagorean theorem $\text{CE} = \sqrt{\text{CD}^{2} + \text{DE}^{2}} = \sqrt{10^{2} + 10^{2}} = 10 \sqrt{2} m$

Therefore, $\text{ME} = 5 \sqrt{2} m$.

Now, in right-angled $\triangle \text{AME}$ right-angled at $\text{M}$, $\text{AM} = \sqrt{\text{AE}^{2} – \text{ME}^{2}} = \sqrt{13^{2} – \left(5 \sqrt{2} \right)^{2}} = \sqrt{169 – 50} = \sqrt{119} m$.

Volume of pyramid = $\frac {1}{3}\text{B}{H} = \frac {1}{3} \times 100 \times \sqrt{119} = \frac {100 \sqrt{119}}{3} m^{3}$.

Ex 3: An architect wants to make a square pyramid and fill it with $12,000$ cubic feet of sand. If the pyramid’s base is $30$ feet on each side, how tall does he need to make it?

Volume of pyramid $\text{V} = 12,000 ft^{3}$.

Side length of square base = $30 ft$.

Area of the base $\text{B} = 30^{2} = 900 ft^{2}$.

Volume of a pyramid $\text{V} = \frac {1}{3} \text{B} \text{H} => \text{H} = \frac {3 \text{V}}{\text{B}} = \frac {3 \times 12,000}{900} = 40 ft$.

Pyramid – A 3D Solid Shape

A pyramid is a three-dimensional shape. A pyramid has a polygonal base and flat triangular faces, which join at a common point called the apex. A pyramid is formed by connecting the bases to an apex. Each edge of the base is connected to the apex and forms the triangular face, called the lateral face. If a pyramid has an $n$-sided base, then it has $n+1$ faces, $n+1$ vertices, and $2n$ edges.

volume of a pyramid

Types of Pyramid

The pyramids are classified depending on the following factors

  • Pyramids based on the type of polygon, of the base
  • Pyramids based on the alignment of the identical bases
  • Pyramids based on the shape of the bases

A pyramid based on the type of the polygon of the base can be of the following two types.

  • Regular Pyramid: If the base of the pyramid is in the shape of a regular polygon, the pyramid is a regular pyramid.
  • Irregular Pyramid: If the base of the pyramid is in the shape of an irregular polygon, the pyramid is an irregular pyramid.
volume of a pyramid

The location of the apex or the top of a pyramid decides whether a pyramid is a right pyramid or an oblique pyramid. 

  • Right Pyramid: A pyramid is named a right pyramid when the location of the apex is exactly over the middle of the base of the pyramid. In other words, when a perpendicular line from the apex intersects the centre of the base, it is a right pyramid.
  • Oblique Pyramid: When the location of the apex is not exactly over the middle but slightly away, then that pyramid is called an oblique pyramid. When it does not intersect the center of the base, it is an oblique pyramid.
volume of a pyramid

There are different types of pyramids based on the shape of their base.

  • Triangular Pyramid: If the base of a pyramid is in the shape of a triangle, it is said to be a triangular pyramid. A triangular pyramid has $6$ edges, $4$ vertices, and $4$ faces. This kind of pyramid can also be called a tetrahedron.
  • Square Pyramid: A square pyramid is formed when the base of the pyramid is in the shape of a square. A square pyramid consists of one square base and three triangular faces. In other words, it has $8$ edges, $5$ vertices, and $5$ faces.
  • Rectangular Pyramid: A rectangular pyramid is formed when the base of the pyramid is in the shape of a rectangle. A rectangular pyramid consists of one rectangular base and three triangular faces. In other words, it has $8$ edges, $5$ vertices, and $5$ faces.
  • Pentagonal Pyramid: A pentagonal pyramid is one that has its base shaped like a pentagon, with the rest of the faces as triangles. This pyramid has $6$ vertices, $10$ edges, and $6$ faces.
volume of a pyramid

Practice Problems

Q 1. Find the volume of the given pyramid.

volume of a pyramid

Q 2. The volume of a $6$-foot-tall square pyramid is $8$ cubic feet. How long are the sides of the base?

Q 3. Find the volume of the given pyramid.

volume of a pyramid

Q 4. Find the volume of the given pyramid.

volume of a pyramid

FAQs

What is the volume of a pyramid?

The amount of space occupied by a pyramid is referred to as the volume of a pyramid. The volume of the pyramid depends on the base area of the pyramid and the height of the pyramid. The unit of volume of the pyramid is expressed in $m^{3}$, $cm^{3}$, $in^{3}$, $ft^{3}$, etc.

What is the formula for the volume of a pyramid?

The formula for the volume of a pyramid is obtained by finding the product of one-third of the base area and height of the pyramid. The volume of a pyramid is given as $\text{V} = \frac {1}{3} \text{B} \times \text{H}$ where, $\text{V}$ is the volume of the pyramid, $\text{B}$ is the area of the base of the pyramid, and $\text{H}$ is the height of the pyramid.

How does the volume of the pyramid change if the type of pyramid changes?

The volume of the pyramid depends on the base area of the pyramid. As the type of pyramid changes, the base of the pyramid changes thereby changing the base area of the pyramid. This change in the base area of the pyramid changes the volume of the pyramid.

Conclusion

The volume of a pyramid is the number of cubic units, occupied by the pyramid completely and is calculated by finding the product of one-third of the base area and the height. Since the formula for finding the area of different polygons are different, so the formulas for calculating the volume of different types of pyramids are different.

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