Pyramid is a polyhedron with a polygonal base and triangles for sides. There are three important parts in any pyramid namely: *base, face and apex*. The base of the pyramid may be of any shape. The faces of the pyramid are mostly isosceles triangle. All the triangular faces meet at a single point called the **apex**.

The

Pyramids are of different types. They are named based on the the base shape of pyramid. The types of pyramid are: square pyramid, triangular pyramid, pentagonal pyramid and hexagonal pyramid.

Related Calculators | |

Pyramid Calculator | Area of a Pyramid Calculator |

Pyramid Volume Calculator | Square Pyramid Volume Calculator |

A Square Pyramid has a square base, 4 triangular faces and an apex :

The**Square Pyramid formulas** are,

Where,

b - base length of the square pyramid.

s - slant height of the square pyramid.

h - height of the square pyramid.

A Triangular Pyramid has a triangular base, 3 triangular faces and an apex

The**Triangular Pyramid formulas** are,

Where,

a - apothem length of the triangular pyramid.

b - base length of the triangular pyramid.

s - slant height of the triangular pyramid.

h - height of the triangular pyramid. A Pentagonal Pyramid has a pentagonal base, 5 triangular faces and an apex

The**Pentagonal Pyramid Formulas** are,

Where,

a - apothem length of the pentagonal pyramid.

b - base length of the pentagonal pyramid.

s - slant height of the pentagonal pyramid.

h - height of the pentagonal pyramid. A Hexagonal Pyramid has a hexagonal base, 6 triangular faces and an apex :

The**Hexagonal Pyramid Formulas** are,

Where,

a - apothem length of the hexagonal pyramid.

b - base length of the hexagonal pyramid.

s - slant height of the hexagonal pyramid.

h - height of the hexagonal pyramid. Some solved problems on pyramid are given below :

### Solved Examples

**Question 1: **Find the base area, surface area and volume of a triangular pyramid of apothem length 3 cm, base length 6 cm, height 10 cm and slant height 12 cm ?

** Solution: **

Given,

a = 3 cm

b = 6 cm

h = 10 cm

s = 12 cm

Base area of a triangular pyramid

= $\frac{1}{2}$ab

= $\frac{1}{2}$ $\times$ 3 cm $\times$ 6 cm

= 9 cm^{2}Surface area of a triangular pyramid

= $\frac{1}{2}$ab + $\frac{3}{2}$bs

= $\frac{1}{2}$ $\times$ (3 cm) $\times$ (6 cm) + $\frac{3}{2}$ $\times$ (6 cm) $\times$ (12 cm)

= 9 cm^{2} + 108 cm^{2}= 117 cm^{2}

Volume of a triangular pyramid

= $\frac{1}{6}$abh

= $\frac{1}{6}$ $\times$ 3 cm $\times$ 6 cm $\times$ 10 cm

= 30 cm^{3}

**Question 2: **Find the base area, surface area and volume of a hexagonal pyramid of apothem length 5 cm, base length 8 cm, height 12 cm and slant height 15 cm ?

** Solution: **

Given,

a = 5 cm

b = 8 cm

h = 12 cm

s = 15 cm

Base area of a hexagonal pyramid

= 3ab

= 3 $\times$ 5 cm $\times$ 8 cm

= 120 cm^{2
}Surface area of a hexagonal pyramid

= 3ab + 3bs

= (3 $\times$ 5 cm $\times$ 8 cm) + (3 $\times$ 8 cm $\times$ 15 cm)

= 120 cm^{2} + 360 cm^{2
}= 480 cm^{2}

Volume of a hexagonal pyramid

= abh

= 5 cm $\times$ 8 cm $\times$ 12 cm

= 480 cm^{3}

The

Where,

b - base length of the square pyramid.

s - slant height of the square pyramid.

h - height of the square pyramid.

A Triangular Pyramid has a triangular base, 3 triangular faces and an apex

The

Where,

a - apothem length of the triangular pyramid.

b - base length of the triangular pyramid.

s - slant height of the triangular pyramid.

h - height of the triangular pyramid. A Pentagonal Pyramid has a pentagonal base, 5 triangular faces and an apex

The

Where,

a - apothem length of the pentagonal pyramid.

b - base length of the pentagonal pyramid.

s - slant height of the pentagonal pyramid.

h - height of the pentagonal pyramid. A Hexagonal Pyramid has a hexagonal base, 6 triangular faces and an apex :

The

Where,

a - apothem length of the hexagonal pyramid.

b - base length of the hexagonal pyramid.

s - slant height of the hexagonal pyramid.

h - height of the hexagonal pyramid. Some solved problems on pyramid are given below :

Given,

a = 3 cm

b = 6 cm

h = 10 cm

s = 12 cm

Base area of a triangular pyramid

= $\frac{1}{2}$ab

= $\frac{1}{2}$ $\times$ 3 cm $\times$ 6 cm

= 9 cm

= $\frac{1}{2}$ab + $\frac{3}{2}$bs

= $\frac{1}{2}$ $\times$ (3 cm) $\times$ (6 cm) + $\frac{3}{2}$ $\times$ (6 cm) $\times$ (12 cm)

= 9 cm

Volume of a triangular pyramid

= $\frac{1}{6}$abh

= $\frac{1}{6}$ $\times$ 3 cm $\times$ 6 cm $\times$ 10 cm

= 30 cm

Given,

a = 5 cm

b = 8 cm

h = 12 cm

s = 15 cm

Base area of a hexagonal pyramid

= 3ab

= 3 $\times$ 5 cm $\times$ 8 cm

= 120 cm

= 3ab + 3bs

= (3 $\times$ 5 cm $\times$ 8 cm) + (3 $\times$ 8 cm $\times$ 15 cm)

= 120 cm

Volume of a hexagonal pyramid

= abh

= 5 cm $\times$ 8 cm $\times$ 12 cm

= 480 cm