# Surface Area of a Pyramid

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

## Formulas

The general formula is:

Surface Area (SA) = ${B+\dfrac{1}{2}Ps}$, here B = base area, P = base perimeter, s = slant height,

Also ${\dfrac{1}{2}Ps}$ = lateral surface area (LSA)

∴ SA = B + LSA

However, there are specific formulas to calculate the surface area of different pyramids. They are given below:

Let us solve some examples involving the above formulas.

## Solved Examples

Calculate the surface area of a triangular pyramid with a base area of 24 cm2, a base perimeter of 12 cm, and a slant height of 16 cm.

Solution:

As we know,
Surface Area (SA) = ${B+\dfrac{1}{2}Ps}$, here B = 24 cm2, P = 12 cm, s = 16 cm
SA = ${24+\dfrac{1}{2}\times 12\times 16}$
= 120 cm2

Finding the surface area of a triangular pyramid when BASE, BASE HEIGHT, and SLANT HEIGHT are known.

Find the total surface area of a regular triangular pyramid with a slant height of 10 cm, base of 6 cm, and a base height of 5.2 cm.

Solution:

As we know,
Total Surface Area (TSA) = ${\dfrac{1}{2}bH+\dfrac{3}{2}bs}$, here b = 6 cm, H = 5.2 cm, s = 10 cm
TSA = ${\dfrac{1}{2}\times 6\times 5.2+\dfrac{3}{2}\times 6\times 10}$
= 285.6 cm2

Find the lateral and total surface area of a square pyramid with a base of 6 cm and a slant height of 7.3 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = 2bs, here b = 6 cm, s = 7.3 cm
LSA = 2 × 6 × 7.3
= 87.6 cm2
Total Surface Area (TSA) = b2 + LSA, here b = 6 cm, LSA = 87.6 cm2
TSA = 62 + 87.6
= 123.6 cm2

Find the total surface area of a rectangular pyramid with a base of 7 cm and 9 cm, and a height of 11 cm.

Solution:

As we know,
Total Surface Area (TSA) = ${lw+\dfrac{1}{2}w\sqrt{4h^{2}+l^{2}}+\dfrac{1}{2}l\sqrt{4h^{2}+w^{2}}}$, here l = 9 cm, w = 7 cm, h = 11 cm
TSA = ${9\times 8+\dfrac{1}{2}\times 7\sqrt{4\times 11^{2}+9^{2}}+\dfrac{1}{2}\times 9\sqrt{4\times 11^{2}+7^{2}}}$
= 250.08 cm2

Find the total surface area of a pentagonal pyramid with a base of 3 cm, apothem of 2.06 cm, and a slant height of 4 cm.

Solution:

As we know,
Total Surface Area (TSA) = ${\dfrac{5}{2}b\left( a+s\right)}$, here b = 3 cm, a = 2.06 cm, s = 4 cm
TSA = ${\dfrac{5}{2}\times 3\times \left( 2.06+4\right)}$
= 45.45 cm2

Find the total surface area of a hexagonal pyramid with a base of 4 cm, apothem of 3.46 cm, and a slant height of 13 cm.

Solution:

As we know,
Total Surface Area (TSA) = 3ab + 3bs, here a = 3.46 cm, b = 4 cm, s = 13 cm
TSA = 3 × 3.46 × 4 + 3 × 4 × 13
= 197.52 cm2