# Surface Area of a Rectangular Pyramid

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

Lateral Surface Area (LSA) = ${\dfrac{1}{2}w\sqrt{4h^{2}+l^{2}}+\dfrac{1}{2}l\sqrt{4h^{2}+w^{2}}}$, here l = base length, w = base width, h = height

∴ Total Surface Area (TSA) = lw + LSA

Let us solve some examples to understand the concept better.

## Solved Examples

Find the lateral and total surface of a rectangular pyramid with a base length of 22 cm, a base width of 13 cm, and a height of 17 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = ${\dfrac{1}{2}w\sqrt{4h^{2}+l^{2}}+\dfrac{1}{2}l\sqrt{4h^{2}+w^{2}}}$, here l = 22 cm, w = 13 cm, h = 17 cm
LSA = ${\dfrac{1}{2}\times 13\sqrt{4\times 17^{2}+22^{2}}+\dfrac{1}{2}\times 22\sqrt{4\times 17^{2}+13^{2}}}$
= 663.63 cm2
Total Surface Area (TSA) = lw + LSA, here l = 22 cm, w = 13 cm, LSA = 663.63 cm2
TSA = 22 × 13 + 663.63
= 949.63 cm2

Find the surface of a right rectangular pyramid with bases of 9 cm and 12 cm, and a height of 11cm.

Solution:

As we know,
Total Surface Area (TSA) = ${lw+\dfrac{1}{2}\times 13\sqrt{4\times 17^{2}+22^{2}}+\dfrac{1}{2}\times 22\sqrt{4\times 17^{2}+13^{2}}}$, here l = 12 cm, w = 9 cm, h = 11 cm
TSA = ${12\times 9+\dfrac{1}{2}\times 9\sqrt{4\times 11^{2}+12^{2}}+\dfrac{1}{2}\times 12\sqrt{4\times 11^{2}+9^{2}}}$
= 363.38 cm2

Let us learn how to calculate the surface area of a rectangular pyramid with slant height.

Finding the surface of a rectangular pyramid when BASE LENGTHBASE WIDTH, and SLANT HEIGHT are known

Find the surface of a rectangular pyramid with bases of 11 cm, 7 cm, and a slant height of 16 cm.

Solution:

Here, we will use the general formula.
Total Surface Area (TSA) = ${B+\dfrac{1}{2}Ps}$, here B = base area, P = base perimeter, l = slant height
B = l × w, here l = 11 cm, w = 7 cm,
= 11 × 7
= 77 cm2
P = 2(l + w), here l = 11 cm, w = 7 cm
= 2 × (11 + 7)
= 36 cm
Plugging the value of B and P in the general formula,
Total Surface Area (TSA) = ${B+\dfrac{1}{2}Ps}$, here B = 77 cm2, P = 36 cm, s = 16 cm
TSA = ${77+\dfrac{1}{2}\times 36\times 16}$
= 365 cm2