Surface Area of a Cone

The surface area (total surface area) of a cone is the entire space occupied by the flat circular base of the cone and its curved surface. The surface area is expressed in square units such as m², cm², mm², and in². Here we will discuss how to find the surface area of a right circular cone.

Formula

The basic formula when slant height and radius are known is:

Derivation

Let us consider a cone with radius r, circumference c, and slant height s.

So, C = 2πr,

And Base Area (B) of the cone = πr²

Now, we unroll the lateral face of the cone first. When placed flat, the lateral face becomes a sector of a circle as shown below. Circumference c becomes the arc length, and s becomes the radius of the sector.

As we know,

Area of a sector (A) ${=\dfrac{1}{2}cs}$, where C = arc length, s = radius,

= ½ × 2πr × s, [∵ C = 2πr]

= πrs = Lateral Surface Area (LSA) of a cone

Here we have derived the formula of LSA.

Now, Surface Area (SA) = LSA + B

∴ SA = LSA + B

= πrs + πr²

Solved Examples

Find the surface area of a cone with a radius of 7 cm and a slant height of 15 cm.

Solution:

As we know,
Surface Area (SA) = πr² + πrs, here r = 7 cm, s = 15 cm, π = 3.141
∴ SA = 3.141 × 7² + 3.141 × 7 × 15
= 483.71 cm²

Find the area of the curved surface of a cone with a radius of 8.5 cm and a slant height of 22 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = πrs, here r = 8.5 cm, s = 22 cm, π = 3.141
∴ LSA = 3.141 × 8.5 × 22
= 587.36 cm²