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

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 + πr^{2}

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) = πr^{2} + πrs, here r = 7 cm, s = 15 cm, π = 3.141 ∴ SA = 3.141 × 7^{2} + 3.141 × 7 × 15 = 483.71 cm^{2}

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 cm^{2}