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

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 cm^{2} Total Surface Area (TSA) = lw + LSA, here l = 22 cm, w = 13 cm, LSA = 663.63 cm^{2} ∴ TSA = 22 × 13 + 663.63 = 949.63 cm^{2}

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 cm^{2}

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 LENGTH, BASE WIDTH, and SLANTHEIGHT 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 cm^{2} 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 cm^{2}, P = 36 cm, s = 16 cm ∴ TSA = ${77+\dfrac{1}{2}\times 36\times 16}$ = 365 cm^{2}