# Volume of Pyramid

A pyramid is a three-dimensional geometric shape that has a polygonal base and triangular faces that meet at a common vertex. The volume of pyramid is the amount of space occupied by the pyramid and is calculated by the formula:

Volume of a Pyramid (V) = (1/3) x Base Area (BA) x Height (H)

Where the base area (BA) is the area of the polygonal base and Height (H) is the perpendicular distance from the base to the apex (top vertex) of the pyramid.

Volume of Pyramid: (Pyramid ):

There are different types of pyramids, depending on their base shapes. If the base is triangle then it is called triangular pyramid. If base shape is rectangle, it is called as rectangular pyramid and so on.

Formulae to calculate different types of pyramids:

Volume of triangular pyramid:
V = 1/3 x Area of triangular base x height (H)
V (volume of pyramid) = 1/3( base x height) / 2 x H Example:

Find the volume of a pyramid with a triangular base, where the base has a length of 10 cm, a height of 8 cm, and a height of pyramid is 12 cm.

Solution: The Base Area of the pyramid is the area of the triangle, which is:

Base Area = (1/2) x Base x Height

= (1/2) x 10  x 8

= 40 cm²

Volume of the triangular pyramid = 1/3 base area x height

= 1/3(40) x 12

Volume of triangular pyramid = 160 cm^3

Volume of rectangular pyramid:
V = 1/3 x Area of rectangular base x height (H)
V (volume of pyramid) = 1/3( length x width)  x H Example:

Find the volume of a rectangular pyramid with a base length of 6 cm, a base width of 4 cm, and a height of 10 cm.

Solution: Using the formula above, we can find the volume of the rectangular pyramid:

V = (lw h)/3

V = (6  x 4  x 10 )/3

V = 80 cm^3

Volume of square pyramid:
V = 1/3 x Area of square base x height(H)
V (volume of pyramid)  = 1/3( length x width)  x H Example:

If the length of each side of a square pyramid is 4 cm and its height is 12 cm. find the volume of square pyramid
Solution:
Area of base = side x side
⇒ Area = 4 x 4
⇒ Area = 16 cm2
Pyramid volume = 1/3 x area of base x height
⇒ Volume = 1/3 x 16 x 12
⇒ Volume = 64 cm3

Volume of Hexagonal pyramid:

V = 1/3 x Area of hexagon x H
V = 6 x (√3 / 4) (side)2 x H
V (volume of pyramid) = 1/3(A x P/ 2) x H ( A = apothem ; P = perimeter) Example:

Find the volume of a regular hexagonal pyramid with an apothem length of 6 cm, side 3 cm and height of the pyramid is 21 cm.
Solution :
apothem = A = 6 cm and side = 3 cm height = H =21 cm
Perimeter = 3 x 6 = 18 cm
Area of Base (hexagon) = A x P /2
⇒ Area = (6 x 18 )/2
⇒ Area = 54 cm2
volume of pyramid=1/3 x( Area of base) x H
∴ V = 1/3 x 54 x 21
∴ Volume = 378 cm3

Calculation of volume of hexagonal pyramid can be explained in other way also, here it is…..

In a hexagonal pyramid, the base is a regular hexagon with sides of length s. To find the area of the base, we can use the formula:

B = (6√3/4)s^2

To find the height h, we need to draw a perpendicular line from the apex to the centre of the base, which divides the hexagonal pyramid into two congruent triangles. The height h is the length of this perpendicular line.

If the hexagonal pyramid has a regular hexagonal base with side length s, and the height from the apex to the centre of the base is c, then the height h can be found using the Pythagorean Theorem:

h^2 = c^2 - (s/2)^2

Now we can plug in these values into the formula for the volume of a pyramid:

V = (1/3)(3√3/2)s^2 × √(c^2 - (s/2)^2)

Simplifying, we get:

V = (√3/2)s^2 × √(4c^2 - s^2)/2

Therefore, the volume of a hexagonal pyramid with base side length s, and height from the apex to the centre of the base c, is

(√3/2)s^2 × √(4c^2 - s^2)/2.

Example: Find the volume of a regular hexagonal pyramid with a base side length of 6 cm and a height of 8 cm.

Solution: First, we need to find the area of the base, using the formula:

B = (3√3/2)s^2

B = (3√3/2)(6 )^2

B = 93.53 cm^2

Next, we need to find the height from the apex to the centre of the base, using the Pythagorean Theorem:

h^2 = c^2 - (s/2)^2

h^2 = (8 )^2 - (6 /2)^2

h^2 = 64 - 9

h^2 = 55

h = √55 cm

Now we can plug in these values into the formula for the volume of a hexagonal pyramid:

V = (√3/2)s^2 × √(4c^2 - s^2)/2

V = (√3/2)(6)^2 × √(4(√55 )^2 - (6 )^2)/2

V = 180√3 cm^3

Therefore, the volume of the hexagonal pyramid is 180√3 cubic cm.

Volume of Pentagonal Pyramid:

Volume = 1/3 x area of pentagon x height (H)
V (volume of pyramid) = 1/3 (A x P/ 2) x H ( A = apothem ; P = perimeter) Example:

Find the volume of a pentagonal pyramid with a base edge length of 5 cm and a height of 10 cm.

Solution: The area of the base is a bit more complicated to find. We can split the pentagon into triangles and use the formula for the area of a triangle:

B = (5 * 5 * sqrt (5) / 4) * 5 = 61.92 square cm

Therefore, the volume is:

V = (1/3) * 61.92 * 10 = 206.4 cubic cm.

We can calculate of volume of pentagonal pyramid in other way also, which is…

V = (1/3)Bh

Where V is the volume, B is the area of the base, and h is the height from the base to the apex.

In a pentagonal pyramid, the base is a regular pentagon with sides of length s. To find the area of the base, we can use the formula:

B = (5/4)s^2 × cot(π/5)

Where cot(π/5) is the cotangent of the interior angle of a regular pentagon, which is approximately 1.3764.

To find the height h, draw a perpendicular line from the apex to the centre of the base, which divides the pentagonal pyramid into two congruent triangles. The height h is the length of this perpendicular line.

If the pentagonal pyramid has a regular pentagonal base with side length s, and the height from the apex to the centre of the base is c, then the height h can be found using the Pythagorean Theorem:

h^2 = c^2 - (s/2)^2

Now we can plug in these values into the formula for the volume of a pyramid:

V = (1/3)(5/4)s^2 × cot(π/5) × √(c^2 - (s/2)^2)

Simplifying, we get:

V = (5/12)s^2 × cot(π/5) × √(4c^2 - s^2)/2

Therefore, the volume of a pentagonal pyramid with base side length s, and height from the apex to the centre of the base c, is

(5/12)s^2 × cot(π/5) × √(4c^2 - s^2)/2.

Example: Find the volume of a regular pentagonal pyramid with a base side length of 5 cm and a height of 10 cm.

Solution: First, we need to find the area of the base, using the formula:

B = (5/4)s^2 × cot(π/5)

B = (5/4)(5 )^2 × cot(π/5)

B = 43.01 cm^2

Next, we need to find the height from the apex to the centre of the base, using the Pythagorean Theorem:

h^2 = c^2 - (s/2)^2

h^2 = (10 )^2 - (5 /2)^2

h^2 = 100 - 6.25

h^2 = 93.75

h = √93.75 cm

Now we can plug in these values into the formula for the volume of a pentagonal pyramid:

V = (5/12)s^2 × cot(π/5) × √(4c^2 - s^2)/2

V = (5/12)(5 )^2 × cot(π/5) × √(4(√93.75 )^2 - (5 )^2)/2

V = 98.01 cm^3

Therefore, the volume of the pentagonal pyramid is 98.01 cubic cm.

Let’s apply these formulas to calculate volume of different pyramids:

Example 1:

Overall, the volume of a pyramid has practical applications in a wide range of fields, from architecture and engineering to agriculture and mathematics.

Find the volume of a pyramid with a square base of side length 6 cm and a height of 8 cm.

Solution: The Base Area of the pyramid is the area of a square, which is: Base Area = side x side = 6 x 6

Base area = 36 cm²

Using the formula, we have:

Volume of Pyramid = (1/3) x Base Area x Height

Volume = (1/3) x 36 x 8

Volume = 96 cm³

Therefore, the volume of the pyramid is 96 cm³

Example 2:
If the length of each side of the base of a triangular pyramid is 6 cm and its height is 10 cm, find the volume of pyramid. ( √3 = 1.73)
Solution:
Area of base = √3 / 4 x (side)2
⇒ Area = √3 / 4 x (6)2
⇒ Area = √3 /4 x 36
∴ Area = 9 √3 cm2
Pyramid volume = 1/3 x area of base x height
⇒ Pyramid volume = 1/3 x 9 √3 x 10
⇒ Volume = 30 √3
⇒ Volume = 30 x 1.73
⇒ Volume = 51.9 cm3Example 3:

A triangular pyramid has a base with sides of 8 cm, 10 cm, and 12 cm, and a height of 9 cm. Find its volume.

Solution: First, find the area of the triangular base using Heron's formula: s = (8 + 10 + 12)/2 = 15

B = √(s(s-8)(s-10)(s-12))

= √(1575*3)

= √1575 ≈ 39.75 cm^2

Then, plug in the values into the formula:

V = (1/3)Bh = (1/3)(39.75)(9) = 119.25 cm^3

Therefore, the volume of the triangular pyramid is 119.25 cubic cms

The volume of a pyramid has many practical applications in various fields. Here are some examples:

Architecture and Engineering: Architects and engineers use the volume of a pyramid to determine the amount of material needed to construct a building or structure. For example, if a pyramid-shaped skylight is being designed for a building, the volume of the pyramid can be calculated to determine the amount of glass needed.

Packaging: Companies that manufacture packaging for products also use the volume of a pyramid to determine the amount of material needed for packaging. For example, a company that manufactures pyramid-shaped tea bags needs to calculate the volume of the pyramid to determine the amount of tea that can be packaged.

Agriculture: The volume of a pyramid is also used in agriculture to calculate the amount of fertilizer needed for a field. For example, if a field is pyramid-shaped, the volume of the pyramid can be calculated to determine the amount of fertilizer needed to cover the field.

Mathematics: The volume of a pyramid is a fundamental concept in mathematics and is used in various mathematical problems and equations. It is also used in geometry to calculate the surface area and volume of complex shapes. Volume :

Volume Formulas
Volume of Irregular Shape
Volume of Cube
Volume of Rectangular Prism(Cuboid)
Volume of Cylinder
Cone volume
Sphere's volume
Hemisphere's volume
Prism's volume
Volume of Pyramid

From Pyramid to Mensuration