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# Surface Area of a Hexagonal Pyramid

The term "pyramid" refers to a three-dimensional object with a polygonal (triangular, square, rectangle, pentagon, hexagon etc) base which is connected with triangular lateral surfaces, all meeting at one point called apex of the pyramid. The number of triangular faces is equal to the number of sides the base has. For example - if the base is a triangle, then the pyramid will have 3 triangular surfaces and if the base is a pentagon, the pyramid has 5 triangular surfaces.

This article with brief you about the hexagonal pyramid. The discussion will especially throw light on the surface area of the hexagonal pyramid. We shall learn what is the formula and method of calculating surface area with the help of solved examples.

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## Definition

A hexagonal pyramid is a pyramid whose base is in the shape of a hexagon. This pyramid has six lateral surfaces in the shapes of the triangle. The height of a hexagonal pyramid is the length of perpendicular drawn from the apex to the base. In a right hexagonal pyramid, the foot of perpendicular lies exactly in the center of the base. A regular right hexagonal pyramid has a regular hexagon as its base and thus all its six triangles are congruent. Recall that a regular hexagon is a six-sided polygon with all equal sides. A right regular hexagon is shown in the following diagram :

The surface area of a hexagonal pyramid is defined as the area of all its surfaces. It is known as the total surface area. It includes the area of base hexagon and area of six triangles. On the other hand, when we consider only the area of triangular surfaces, not the area of the base, it is known as the lateral surface area.

## Formula

The formula for finding the surface area (total surface area) of a hexagonal pyramid can be derived in the following way.

When a right regular hexagon is opened, it looks similar to the following image,

Suppose that the side of the hexagon is represented by "s" and apothem (perpendicular drawn from the center to any of side) be "a". Assume that "l" be the slant height (height of the triangle) of the pyramid.

Then, the surface area is given by :

Surface area = Area of six triangles + Area of base

Surface area =
6 $\times$ $\frac{1}{2}$ base $\times$ slant height + $\frac{1}{2}$ perimeter $\times$ apothem

TSA = 6 $\times$ $\frac{1}{2}$ $\times$ s $\times$ l + $\frac{1}{2}$ $\times$ 6s $\times$ a

TSA = 3 s l + 3 a s = 3s (l + a)

Also, lateral surface area is -

LSA = 6
$\times$ $\frac{1}{2}$ base $\times$ slant height

LSA = 3 s l

## How To Find Surface Area of Hexagonal Pyramid

In order to find the surface area of a right hexagonal pyramid, we should follow the steps mentioned below.

Step 1 : Find the apothem "a"of a hexagon. When side "s" is given, the apothem can be calculated by a = $\frac{s}{2\ tan\ 30^{\circ}}$.

Step 2 : Determine slant height "l".

Step 3 : Substitute all three values in the formula to find the surface area.

## Examples

The examples of calculating the surface area of the hexagonal pyramid are given below.

Example 1 : Find the surface area of a hexagonal pyramid whose side is 10 cm and slant height of the pyramid is 14 cm.

Solution :
Side of base, s = 10 cm

Apothem, a = $\frac{s}{2\ tan\ 30^{\circ}}$

a = $\frac{10}{2\ \frac{1}{\sqrt{3}}}$

a = 5 $\sqrt{3}$

Slant height of pyramid, l = 14 cm

Surface area = 3 s l + 3 a s

Surface area = 3 $\times$ 10 $\times$ 14 + 3 $\times$ 5 $\sqrt{3}$ $\times$ 10

= 420 + 150 $\sqrt{3}$

= 420 + 259.81 = 679.81 cm$^{2}$
Example 2 : Calculate the slant height and side of the base of a hexagonal pyramid if they are equal and lateral surface area of the pyramid is 42 square inches.

Solution : Given that LSA = 42 sq in

And slant height = side of base

or l = s

Formula for lateral surface area is

LSA = 3 s l

42 = 3 s s

42 = 3s$^{2}$

s$^{2}$ = 14

s = $\sqrt{14}$ $\approx$ 3.74 inches

Thus, slant height = side of base = 3.74 inches
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