# Hexagonal Pyramid

## Definition

A hexagonal pyramid is a solid with a hexagonal base bounded by six lateral faces meeting at a common point, known as the apex. The lateral faces are in shape of isosceles triangles.

How many faces, edges and vertices does a hexagonal pyramid have?

A hexagonal pyramid has 7 faces, 7 vertices, and 12 edges. Since it has 7 flat faces, a hexagonal pyramid is also called a heptahedron.

Like all other polyhedrons, we can calculate the surface area and the volume of a hexagonal pyramid.

## Formulas

### Volume

The formula is:

Volume (V) = ${\dfrac{\sqrt{3}}{2}b^{2}h}$, here b = base, h = height

Let us solve an example to understand the concept better.

Find the volume of a hexagonal pyramid with a base of 7 cm, and a height of 10 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{\sqrt{3}}{2}b^{2}h}$, here b = 7 cm, h = 10 cm
V = ${\dfrac{\sqrt{3}}{2}\times 7^{2}\times 10}$
= 424.35 cm3

### Surface Area

We can calculate 2 types of surface areas: (1) Lateral Surface Area (LSA), and (2) Total Surface Area (TSA)

The formula is:

Surface Area (SA) = 3ab + 3bs, here a = apothem, b = base, s = slant height,

Also, 3bs = lateral surface area (LSA)

∴ SA = 3ab LSA

Let us solve some examples to understand the concept better.

Find the surface area of a hexagonal pyramid with a base of 9 in, an apothem of 7.79 in, and a slant height of 16 in.

Solution:

As we know,
Total Surface Area (TSA) = 3ab + 3bs, here a = 7.79 in, b = 9 in, s = 16 in
TSA = 3 × 7.79 × 9 + 3 × 9 × 16
= 642.33 in2

Find the lateral and total surface areas of a hexagonal pyramid with a base of 3 cm, an apothem of 2.59 cm, and a slant height of 7 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = 3bs, here b = 3 cm, s = 7 cm
LSA = 3 × 3 × 7
= 63 cm2
Total Surface Area (TSA) = 3ab + LSA, here a = 2.59 cm, b = 3 cm, LSA = 63 cm2
TSA = 3 × 7.79 × 9 + 63
= 273.33 cm2