# Surface Area of a Triangular Pyramid

The surface area, or total surface area (TSA), of a triangular pyramid, is the entire space occupied by its four 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 triangular pyramid also includes its lateral surface area (LSA).

Lateral Surface Area (LSA) = ${\dfrac{1}{2}Ps}$, here P = base perimeter, s = slant height

Total Surface Area (TSA) = B + LSA

Let us solve some example to understand the above concept better.

## Solved Examples

Find the surface area of a regular triangular pyramid with a base area of 43 cm2, a base perimeter of 30 cm, and a slant height of 8 cm.

Solution:

As we know,
Total Surface Area (TSA) = ${B+\dfrac{1}{2}Ps}$, here B = 43 cm2, P = 30 cm, s = 8 cm
TSA = ${43+\dfrac{1}{2}\times 30\times 8}$,
= 163 cm2

Calculate the lateral and total surface area of a triangular-based pyramid with a base of 20 cm, a base height of 12 cm, and a slant height of 19 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = ${\dfrac{1}{2}Ps}$,
P = 3 × b, here b = 20 cm
LSA = ${\dfrac{3}{2}bs}$, here b = 20 cm, s = 14 cm
= ${\dfrac{3}{2}\times 20\times 14}$
= 570 cm2
Total Surface Area (TSA) = B + LSA
Now, B = ${\dfrac{1}{2}bH}$, here b = 20 cm, H = 12 cm
∴ ${TSA=\dfrac{1}{2}bH+LSA}$, here b = 20 cm, H = 12 cm, LSA = 570 cm2
∴ ${TSA=\dfrac{1}{2}\times 20\times 12+570}$
= 690 cm2

Finding the surface area of a regular (equilateral) triangular pyramid when the BASE and SLANT HEIGHT are known

Find the surface area of a regular triangular pyramid with a base of 16 cm and a slant height of 19 cm.

Solution:

Here, we will use an alternative formula.
Total Surface Area (TSA) = ${\dfrac{\sqrt{3}}{4}b^{2}+\dfrac{1}{2}\times 3bs}$, here b = 16 cm, s = 19 cm
TSA = ${\dfrac{\sqrt{3}}{4}\times 16^{2}+\dfrac{1}{2}\times 3\times 16\times 19}$
= 566.85 cm2