# Surface Area of a Pentagonal Pyramid

The surface area, or total surface area (TSA), of a pentagonal pyramid, is the entire space occupied by its six faces. It is measured in square units such as m2, cm2, mm2, and in2.

## Formula

The formula is:

The formula to calculate the surface area of a pentagonal pyramid also includes its lateral surface area (LSA).

Lateral Surface Area (LSA) = ${\dfrac{5}{2}bs}$, here b = base, s = slant height

Total Surface Area (TSA) = ${\dfrac{5}{2}ab+LSA}$

Let us solve some examples to understand the concept better.

## Solved Examples

Find the lateral and total surface area of a pentagonal pyramid with an apothem of 2.75 cm, a  base of 4 cm, and a slant height of 6.4 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = ${\dfrac{5}{2}bs}$, here b = 4 cm, s = 6.4 cm
LSA = ${\dfrac{5}{2}\times 4\times 6.4}$
= 64 cm2
Total Surface Area (TSA) = ${\dfrac{5}{2}ab+LSA}$, here a = 2.75 cm, b = 4 cm, LSA = 64 cm2
TSA = ${\dfrac{5}{2}\times 2.75\times 4+64}$
=91.5 cm

Find the surface area of a pentagonal pyramid with an apothem of 5.16 cm, a  base of 7.5 cm, and a slant height of 9 cm.

Solution:

As we know,
Total Surface Area (TSA) = ${\dfrac{5}{2}b\left(a+s\right)}$, here a = 5.16 cm, b = 7.5 cm, s = 9 cm
TSA =  ${\dfrac{5}{2}\times 7.5\times \left(5.16+9\right)}$
= 265.5 cm2

Finding the surface area of a pentagonal pyramid when the BASE and HEIGHT are known

Find the surface area of a pentagonal pyramid with a base of 6 cm and a height of 7 cm.

Solution:

Here, we will use an alternative formula.
Total Surface Area (TSA) = ${1.73b^{2}+\dfrac{5}{2}bh}$, here b = 6 cm, h = 7 cm
TSA =  ${1.73\times 6^{2}+\dfrac{5}{2}\times 6\times 7}$
= 62.28 + 105
= 167.28 cm2