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:
Surface Area of a Triangular Pyramid
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 SLANTHEIGHT are known
Find the surface area of a regulartriangular 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
