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Volume of Pyramid – Explanation & Examples

A pyramid is a 3-dimensional diagram whose polygonal base is connected to the apex by triangular faces in geometry. The triangular faces of a pyramid are known as lateral faces, and the perpendicular distance from the apex (vertex) to the base of a pyramid is known as the height.

Pyramids are named after the shape of their bases. For instance, a rectangular pyramid has a rectangular base, a triangular pyramid has a triangular base, a pentagonal pyramid has a pentagonal base, etc.

How to Find the Volume of a Pyramid?

In this article, we discuss how to find the volume of pyramids with different types of bases and solve word problems involving a pyramid’s volume.

The volume of a pyramid is defined as the number of cubic units occupied by the pyramid. As stated before, the name of a pyramid is derived from the shape of its base. Therefore, the volume of a pyramid also depends on the shape of the base.

To find the pyramid’s volume, you only need the dimensions of the base and the height.

Volume of a pyramid formula

The general volume of a pyramid formula is given as:

Volume of a pyramid = 1/3 x base area x height.

V= 1/3 Ab h

Where Ab = area of the polygonal base and h = height of the pyramid.

Note: The volume of a pyramid varies slightly depending on the polygonal base.

Example 1

Calculate the volume of a rectangular pyramid whose base is 8 cm by 6 cm and the height is 10 cm.

Solution

For a rectangular pyramid, the base is a rectangle.

Area of a rectangle = l x w

= 8 x 6

= 48 cm2.

And by the volume of a pyramid formula, we have,

Volume of a pyramid = 1/3Abh

= 1/3 x 48 cm2 x 10 cm

= 160 cm3.

Example 2

The volume of a pyramid is 80 mm3. If the pyramid’s base is a rectangle that is 8 mm long and 6 mm wide, find the pyramid’s height.

Solution

Volume of a pyramid = 1/3Abh

⇒ 80 = 1/3 x (8 x 6) x h

⇒ 80 = 15.9h

By dividing both sides by 15.9, we get,

h = 5

Thus, the height of the pyramid is 5 mm.

Volume of a square pyramid

To obtain the formula for the volume of a square pyramid, we substitute the base area (Ab) with the area of a square (Area of a square = a2)

Therefore, the volume of a square pyramid is given as:

Volume of a square pyramid = 1/3 x a2 x h

V = 1/3 a2 h

Where a = side length of the base (a square) and h = height of the pyramid.

Example 3

A square pyramid has a base length of 13 cm and a height of 20 cm. Find the volume of the pyramid.

Solution

Given:

Length of the base, a = 13 cm

height = 20 cm

Volume of a square pyramid = 1/3 a2 h

By substitution, we have,

Volume = 1/3 x 13 x 13 x 20

= 1126.7 cm3

Example 4

The volume of a square pyramid is 625 cubic feet. If the height of the pyramid is 10 feet, what are the dimensions of the pyramid’s base?

Solution

Given:

Volume = 625 cubic feet.

height = 10 feet

By the volume of a square formula,

⇒ 625 = 1/3 a2 h

⇒ 625 = 1/3 x a2 x 10

⇒ 625 = 3.3a2

⇒ a2 =187.5

⇒ a = = √187.5

a =13.7 feet

So, the dimensions of the base will be 13.7 feet by 13.7 feet.

Example 5

The base length of a square pyramid is twice the height of the pyramid. Find the dimensions of the pyramid if it has a volume of 48 cubic yards.

Solution

Let the height of the pyramid = x

the length = 3x

volume = 48 cubic yards

But, the volume of a square pyramid = 1/3 a2 h

Substitute.

⇒ 48 = 1/3 (3x)2 (x)

⇒ 48 = 1/3 (9x3)

⇒ 48 = 3x3

Divide both sides by 3 to get,

⇒ x3 =16

⇒ x = 3√16

x = 2.52

Therefore, the height of the pyramid = x ⇒2.53 yards,

and each side of the base is 7.56 yards

Volume of a trapezoidal pyramid

A trapezoidal pyramid is a pyramid whose base is a trapezium or a trapezoid.

Since we know, area of a trapezoid = h1 (b1 + b2)/2

Where h = height of the trapezoid

b1 and b2 are the lengths of the two parallel sides of a trapezoid.

Given the general formula for the volume of a pyramid, we can derive the formula for the volume of a trapezoidal pyramid as:

Volume of a trapezoidal pyramid = 1/6 [h1 (b1 + b2)] H

Note: When using this formula, always remember that h is the height of the trapezoidal base and H is the height of the pyramid.

Example 6

The base of a pyramid is a trapezoid with parallel sides of length 5 m and 8 m and a height of 6 m. If the pyramid has a height of 15 m, find the volume of the pyramid.

Solution

Given;

h = 6 m, H = 15 m, b1 =5 m and b2 = 8 m

Volume of a trapezoidal pyramid = 1/6 [h1 (b1 + b2)] h

= 1/6 x 6 x 15 (5 + 8)

= 15 x 13

=195 m3.

Volume of a triangular pyramid

As we know, the area of a triangle;

Area of a triangle = 1/2 b h

Volume of a triangular pyramid = 1/3 (1/2 b h) H

Where b and h are the base length and height of the triangle. H is the height of the pyramid.

Example 7

Find the area of a triangular pyramid whose base area is 144 in2 and the height is 18 in.

Solution

Given:

Base area = 144 in2

H = 18 in.

Volume of a triangular pyramid = 1/3 (1/2 b h) H

= 1/3 x 144 x 18

= 864 in3

Practice Problems

  1. What is the volume of a 12 units high pyramid with a rectangular base measuring 8 units by 9 units?
  2. Consider a pyramid with an isosceles triangle base having two sides of length 14 units each and 16 units. Find the volume of the pyramid if its height is 22 units.
  3. Consider a pyramid with a square base of 11 cm each. If the volume of this pyramid is 520 cm3, what is the height of this pyramid?

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