# Lateral Surface Area of a Prism

The lateral surface area of a prism is the total of the area of the faces.

$\therefore$Lateral Surface Area $= h\left( {AB} \right) + h\left( {BC} \right) + h\left( {CD} \right) + h\left( {DA} \right)$

Lateral Surface Area $= h\left( {AB + BC + CD + DA} \right)$

Lateral Surface Area $=$ Perimeter of the base $\times$ height of the prism

Lateral Surface Area $=$ Perimeter of the base times the altitude

Rule 1: The lateral area of the prism is equal to the perimeter of the base times the altitude.

Rule 2: The total surface area of a prism is the sum of the lateral areas and the area of its base.

Example:

Find the area of the whole surface of a right triangular prism whose height is $36$m and the sides of whose bases are $51,37$ and $20$m, respectively.

Solution:

In all there are five plane figures, i.e., two triangles and three rectangles. Since both the rectangles are of equal area,

Area of both triangles $= \sqrt {s\left( {s – a} \right)\left( {s – b} \right)\left( {s – c} \right)}$

Area of both triangles $= \sqrt {54\left( {54 – 51} \right)\left( {54 – 20} \right)\left( {54 – 37} \right)}$

Area of both triangles $= \sqrt {54 \times 3 \times 34 \times 17} = 612$ square meters

Area of all three rectangles   $= 36\left( {51 + 20 + 37} \right) = 36 \times 108 = 3888$ square meters

Area of the whole surface $= 3888 + 612 = 4500$ square meters