Lateral area of a solid is sum of surface areas of all lateral faces. Lateral area does not include area of base or top. It is also referred as lateral surface area.

The SI unit of lateral area can be calculated as unit2 or square unit. There are different formulas for calculating lateral area of different shapes.

## Lateral Area of a Pyramid

Lateral area of a pyramid is sum of surface areas all the slant triangles. In lateral surface area, the area of base is not included. The base of pyramid can be any polygon. A right regular pyramid or a right pyramid is one which has regular polygon as its base. Slant height of a pyramid is distance between apex of lateral triangle to its base. Following figure demonstrates a regular pyramid:

The formula for lateral area of a pyramid is given below:
Where,
P = Perimeter of base
s = Slant height.

## Lateral Area of a Cylinder

Lateral area of cylinder is surface area of lateral surface of cylinder excluding areas of top and bottom. Height of a cylinder is perpendicular drawn from the center of the top face to the bottom face. Right circular cylinder or simply a right cylinder is a cylinder in which line segment joining centers of two end faces is perpendicular to the base. Lateral area of cylinder is also known is curved surface area.

If "r" be the radius of base and "h" be the height of a right cylinder, then formula for its lateral surface area is given below:

## Lateral Area of a Prism

Lateral area or lateral surface area of a prism is surface area of a prism excluding areas of top and bottom. End faces of prism can be any polygon. Height of a prism is perpendicular drawn from the center of the top face to the bottom face.

Right prism
is a prism whose lateral edges are perpendicular to the bases.

The formula for lateral surface area of a prism is given below:
Where,
P = Perimeter of base
h = Height of prism.

## Lateral Area of a Cone

Lateral surface area or lateral area of cone is the area of the lateral portion which does not include area of base. Lateral area of cone is also known as curved surface area. Height of a cone is perpendicular drawn from apex of the cone to its base. Right circular cone or simply right cone is a cone in which line segment joining centers of two end faces is perpendicular to the base. Slant height of a cone is height of lateral surface from apex to base and denoted by "l".
Slant height is given by the following formula (using Pythagorean theorem):
$l=\sqrt{r^{2}+h^{2}}$
Where,
h = Height.

The formula for lateral surface area of a cone is given below:
Where,
l = Slant height

## Lateral Area of a Sphere

Lateral surface area of a sphere is the area of its outer surface.

Formula for lateral area of a sphere is given below:
Where r = Radius of sphere.

## Lateral Area of Cube

Lateral area of cube is the surface area of all four lateral faces. Lateral area of a cubical room or cubical box is the area of 4 walls. It does not include area of top and bottom.

If all the edges of a cube measure "a", then the formula for lateral area of a cube is given below:

## Finding Lateral Area

Few problems based on lateral area are as follows:

### Solved Examples

Question 1: Find the lateral surface area of a spherical ball whose radius is 7 cm.
Solution:

$Lateral\ Area\ of\ sphere = 4\pi r^{2}$

$Lateral\ Area\ of\ sphere = 4$ $\times$ $\frac{22}{7}$ $\times$ $7^{2}$

= 616

The lateral surface area of a spherical ball is 616 cm2

Question 2: A manufacturing company wishes to manufacture a conical tent with radius 6 m and height 8 m. What is the cost of canvas at the rate of Rs. 10 per m2?
Solution:

Height (h) = 8 m
Slant height is given by the following formula:
$l = \sqrt{r^{2} + h^{2}}$
$l = \sqrt{6^{2} + 8^{2}}$
= 10
Slant height of cone = 10 m
$Lateral\ area\ of\ cone = \pi r l$

= $\frac{22}{7}$ $\times$ 6 $\times$ 10$= 188.57 So the lateral area of cone = 188.57 m2 Now, Cost of canvas = Rs. 10 per m2 Cost of canvas in manufacturing of tent = 188.571$\times$10 = 1885.71 Therefore, the cost of canvas in manufacturing of tent is Rs. 1885.71. Question 3: Find the lateral surface area of a cylinder of radius 7 cm and height 20 cm. Solution: Radius r = 7 cm Height h = 20 cm Lateral surface area of a cylinder =$2\pi rh$= 2$\times\frac{22}{7}\times$7$\times$20 = 880$\therefore\$ The lateral surface area of a cylinder is 880 cm2