Table of Contents

Last modified on March 28th, 2023

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 m^{2}, cm^{2}, mm^{2}, and in^{2}.

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.

**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 cm^{2}

Total Surface Area (*TSA*) = ${\dfrac{5}{2}ab+LSA}$*, *here a = 2.75 cm, b = 4 cm, LSA = 64 cm^{2}

∴*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 cm^{2}

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 cm^{2}

Last modified on March 28th, 2023