# 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