A cone is a 3-dimensional solid object that has a circular base and one vertex. Cone is a three-dimensional geometric shape that formed by the locus of all straight line segments to the apex. The surface area of the cone equals the area of the circle plus the area of the cone. Surface area of a cone depends on whether cone is closed or open.

The surface area of a cone whose circular base has radius 'r' and whose slant height is 'l' is SA = $\pi$ r(r + l).

Formula for the surface area of a cone:

Cone

Formula:

Surface Area of a Cone = $\pi$ r(r + l)

Here, l = $\sqrt{ r^2 + h^2}$

Where,
r = Radius of a cone
h = Height of a cone
l = Slant height of a cone.

The surface area of the cone equals the area of the circle plus the area of the cone. To find the surface area of a cone, find the area of the circle and area of the cone.

Steps for finding the Surface Area of a Cone:

Step 1:
Measure the radius or slant height of the cone.(If not given)

Step 2:
Find the area of the circle or base of the cone (using formula A = $\pi r^2$).

Step 3:
Find the area of a cone($\pi$ r l) by measuring its slant height. (Make sure you use the same form of measurement as the radius).

Step 4:
Write answer in the proper square unit of measurement.


Example:

The height of a circular cone is 15 inches and the radius of the base is 20 inches. What is the surface area of the cone ?

Solution:
Given

Height of a cone (h) = 15 inches
Radius of the cone (r) = 20 inches
Surface area of the cone (SA) = ?

Step 1:

Find the slant height of the cone:

We know that, l2 = r2 + h2

=> l2 = (20)2 + (15)2

=> l2 = 400 + 225

=> l2 = 625

or l = 25

Step 2:

Area of the circle:

Area of the circle = $\pi r^2$

= $\pi$ (20)2

= 400 $\pi$

=> Area of the circle = 400 $\pi$

Step 3:

Find the area of a cone:

Area of the cone = $\pi$ r l

=> Area of the cone = $\pi$ * 20 * 25

= 500 $\pi$

=> Area of the cone = 500 $\pi$

Step 4:

Surface Area of a Cone = $\pi$ r(r + l)

or Surface Area of a Cone = $\pi$ r2 + $\pi$ rl

= 400 $\pi$ + 500$\pi$

= (400 + 500)$\pi$

= 900 $\pi$

Hence the surface area of a cone = 900 $\pi$ inches2 .
The lateral surface area of a cone is the area of the lateral or side surface. If a cone of radius 'r' and slant height 'l' then the lateral surface area of a cone is $\pi$ rl.

Formula:

Lateral Surface Area of a Cone (LSA) = $\pi$ r l


Where, 'r' is the radius and 'l' is the slant height of a cone.


Example:

A circular cone is 12 yard high and the diameter of the base is 14 yard. What is the lateral surface area of the cone?

Solution:
Step 1:

Slant height of a cone (l) = 12 yard
Diameter of a cone = 14 yard
Lateral surface area of a cone = ?

Step 2:

Find the radius of a cone:
Radius of a cone = $\frac{Diameter}{2}$ = $\frac{14}{2}$

=> Radius of a cone (r) = 7 yard

Step 3:


Lateral Surface Area of a Cone (LSA) = $\pi$ r l

=> LSA = $\frac{22}{7}$ * 7 * 12

= 264

Hence the lateral surface area of the cone is 264 yd2 .


The total surface area of a cone is the sum of the area of its base and the lateral or side surface. The lateral surface area of a cone is the area of the lateral or side surface only. If a cone of radius 'r' and slant height 'l' is cut along the slant height and opened out flat then radius of the sector formed is 'l' and the arc length AB is circumference of the circle (2$\pi$r).

Area of Cone

Total surface area of the cone = Area of the base + Area of curved surface

=> TSA = $\pi r^2$ + $\pi$rl

=> TSA = $\pi$ r(r + l)

Formula:

Total Surface Area of a Cone ( TSA) = $\pi$ r(r + l)


Where,
'r' is the radius and 'l' is the slant height of a cone.


Example:

Find the total surface area of a cone if its slant height is four times the radius, and base diameter of the cone is 10 cm.

Solution:
Given

Diameter of the cone = 10 cm
Slant height of the cone = 4(Radius)

Step 1:

Radius of the cone (r) = $\frac{10}{2}$ = 5 cm
Slant height of the cone (l) = 4 * 5 = 20 cm

Step 2
:
Total Surface Area of a Cone ( TSA) = $\pi$ r(r + l)

=> TSA = $\frac{22}{7}$ * 5(5 + 20)

= $\frac{22}{7}$ * 5 * 25

= 392.85

Hence the total surface area of a cone is 392.85 cm2 .
Frustum of a cone formed from a cone with a circular base by cutting off the tip of the cone with a cut perpendicular to the height, forming a lower base and an upper base that are circular and parallel to each other. If r and R are the radii of ends, h is height and 'l' is the slant height of a cone, the surface area of frustum of a cone is given below:

Frustum of a Cone

Formula:

Surface Area of Frustum of a Cone = $\pi$(R + r)$\sqrt{(R - r)^2 + h^2}$


Where, r and R are the radii of ends, h is height and 'l' is the slant height of frustum of a cone.