Surface Area of Cone
The surface area of a cone is the amount of area occupied by the surface of a cone. A cone is a 3D shape that has a circular base. This means the base is made up of a radius or diameter. The distance between the center of the base and the topmost part of the cone (of course, in the case of ice cream, this portion is at the bottom) is the height of the cone. We can find the surface area of the cone in two ways  total surface area and curved surface area of cone.
In this article, we will learn how to calculate the surface area of a cone in this article. The total surface area includes both curved and flat circular area whereas the curved surface area includes the area of only the curved surface. We will go through the formula and solved a few examples for a better understanding of the concept.
1.  What is the Surface Area of Cone? 
2.  Surface Area of Cone Formula 
3.  Curved Surface Area of Cone 
4.  Derivation of Surface Area of Cone 
5.  FAQs on Surface Area of Cone 
What is the Surface Area of Cone?
The area occupied by the surface/boundary of a cone is known as the surface area of a cone. It is always measured in square units. Stacking many triangles and rotating them around an axis gives the shape of a cone. As it has a flat base, thus it has a total surface area as well as a curved surface area. We can classify a cone as a right circular cone or an oblique cone. The vertex in the right circular cone is usually vertically above the center of the base whereas the vertex of the cone in an oblique cone is not vertically above the center of the base.
Surface Area of Cone Formula
As a cone has a curved surface, thus we can express its curved surface area as well as total surface area. A cone has two kinds of surface area:
 Total Surface Area
 Curved Surface Area
If the radius of the base of the cone is "r" and the slant height of the cone is "l", the surface area of a cone is given as:
 Total Surface Area, T = πr(r + l) square units
 Curved Surface Area, S = πrl square units
By applying Pythagoras theorem on the cone, we can find the relation between the surface area of the cone and its height. We know, h^{2} + r^{2} = l^{2} where h is the height of the cone, r is the radius of the base, and l is the slant height of the cone.
⇒ l = √(h^{2} + r^{2})
Thus,
 The total surface area in terms of height can be given as, T = πr(r + l) = T = πr(r + √(h^{2} + r^{2})).
 The curved surface area of the cone in terms of height can be given as S = πrl = πr(√(h^{2} + r^{2})).
Curved Surface Area of Cone
We discussed the formula to find the curved surface area of cone, let us now understand its meaning. Like other threedimensional shapes, the cone also has both flat and curved surfaces. The curved surface area of cone refers only to the curved part of the cone which is other than the circular flat base. To find the curved surface area of cone, we multiply the radius and slant height of the cone by pi(π). Let us derive the formula for the curved surface area of cone below.
Curved Surface Area of Cone Formula
The curved surface area of the cone can be given by finding the area of the sector by using the formula,
Area of the sector (in terms of length of arc) = (arc length × radius)/ 2 = ((2πr) × l)/2 = πrl.
∴ The curved surface area of a cone, S = πrl units^{2}.
Derivation of Surface Area of Cone
Let us take a cone of height "h", base radius "r", and slant height "l". In order to determine the surface area of cone derivation, we cut the cone open from the center which looks like a sector of a circle (a plane shape).
The total surface area of cone = area of the base of cone + curved surface area of a cone
⇒ Total surface area of cone = πr^{2} + πrl = πr (r + l).
∴ The total surface area of cone, T = πr (r + l) units^{2}
Finding Surface Area of Cone
Let us solve an example below to understand the application of the total surface area of cone and the curved surface area of cone.
Example: Find the total surface area and curved surface area of the cone whose radius is 7 inches and slant height is 3 inches. (Use π = 22/7).
We know, the total surface area of the cone is πr (r + l), and the lateral surface area of a cone is πrl. Given that: r = 7 inches, l = 3 inches, and π = 22/7. Thus, total surface area of cone, T = πr (r + l) = (22/7) × 7 × (7 + 3) = (22/7) × 7 × 10 = 22 × 10 = 220 in^{2}.
∴ The total surface area of the cone is 220 in^{2}.
The curved surface area of the cone, S = πrl = (22/7) × 7 × 3 = 66 in^{2}. ∴ The curved surface area of the cone is 66 in^{2}.
Let us look at some more examples of the surface area of a cone for a deeper understanding.
Surface Area of Cone Examples

Example 1: What is the slant height of the cone if the total surface area of the cone is 616 in^{2} and the radius is 7 inches?
Solution: The given dimensions are, the total surface area of the cone = 616 in^{2 }and the radius of the cone = 7 inches. Let the slant height = x inches.
Substituting the values in the surface area of the cone formula,
Total surface area of cone = πr (r + l) = (22/7) × 7 × (7 + x) = 616
⇒ 22 × (7 + x) = 616
⇒ 7 + x = 28
⇒ x = 21 inches
Answer: The slant height of cone is 21 inches.

Example 2: What is the height of cone whose radius is 7 inches and curved surface area is 550 in^{2} . (Use π = 22/7)
Solution: The given dimensions are, radius of cone = 7 in and curved surface area = 550 in^{2}. Let the value of slant height be "l" and height of cone be "h".
Substituting the values in the curved surface area of the cone formula,
πrl = (22/7) × 7 × l = 550 in^{2}
⇒ 22 × l = 550
⇒ l = 550/22 = 25 inches
l = √(h^{2} + r^{2})
⇒ h = √(l^{2}  r^{2}) = √(25^{2}  7^{2}) = √576 = 24 inches
Answer: The height of cone is 24 inches.

Example 3: Find the total surface area of a cone with radius 14 units and slant height 8 units. (Use π = 22/7)
Solution: The given dimensions are, radius of cone (r) = 14 units and slant height (l) = 8 units.
Substituting the values in the total surface area of the cone formula,
πr (r + l) = (22/7) × 14 × (14 + 8)
= 22 × 2 × 22
= 968 square units
Answer: The total surface area of the cone is 968 square units.
FAQs on Surface Area of Cone
What is the Surface Area of a Cone?
The measure of the area occupied by the surface of a cone is referred to as the surface area of the cone. There are two types of the surface area of the cone that can be categorized as the total surface area and the curved surface area of the cone.
How to Find Surface Area of Cone?
The surface area of a cone can be found by using the following steps:
 Step 1: Identify the given values of radius, slant height, and height of the cone.
 Step 2: Use the appropriate formula to calculate the surface area. If the total surface area of the cone has to be found, use the formula πr (r + l) square units, and if the curved surface area of the cone has to be found, use the formula πrl square units.
 Step 3: Simplify and write the answer in square units.
What is the Curved Surface Area of Cone?
The area of the curved surface of a cone is known as the curved surface area of the cone. The formula to calculate the curved surface area of a cone is πrl where "r" is the radius of the base and "l" is the slant height of the cone. In this, we do not consider the area of the base of the cone which is in the shape of a circle.
What is the Curved Surface Area of a Right Circular Cone?
A right circular cone is defined as the cone whose axis is the line joining the vertex and the midpoint of the circular base. Thus, the curved surface area of a right circular cone is given as πrl where "r" is the radius of the base and "l" is the slant height. In terms of height, the curved surface area of a right circular cone is given as πr(√(h^{2} + r^{2})) where "h" is the height of the right circular cone.
How Do You Find the Surface Area of Cone with Slant Height and Diameter?
The total surface area of a cone with slant height and diameter of the cone can be found by using the formula, T = π(D/2) ((D/2) + l), where D is the diameter and l is the slant height. The curved surface area of the cone with slant height and diameter can be found by using the formula S = π(D/2)l, where D is the diameter and l is the slant height.
How do you Find the Total Surface Area of a Cone?
The total surface area of a cone can be found by using the below steps:
 Step 1: Check the values given in the question.
 Step 2: Substitute the values of radius and slant height in the formula πr (r + l). In case, if there is no slant height given, we write the value of slant height in terms of the height of the cone, "h", by substituting l = √(h^{2} + r^{2}) which gives the value, T = πr(r + √(h^{2} + r^{2})) where "r" is the radius of the cone and "h" is the height of the cone.
 Step 3: Now, find the value of the total surface area.
 Step 4: Write the final answer in terms of square units.
How do you Find the Curved Surface Area of a Cone?
The curved surface area of a cone can be found by using the steps given below:
 Step 1: Write the values given in the question.
 Step 2: Substitute the values of radius and slant height of the given question in the formula πrl. In case, if there is no slant height given, we write the value of slant height in terms of the height of the cone, "h", by substituting l = √(h^{2} + r^{2}) which gives the value, S = πr√(h^{2} + r^{2}) where "r" is the radius of the cone and "h" is the height of the cone.
 Step 3: Now, find the value of the curved surface area and write the answer in terms of square units.
What Happens to the Surface Area of a Cone if the Slant Height and the Radius of the Base are Doubled?
The surface area of the cone depends on the radius of the base and its slant height. Thus, the total surface area and curved surface area of the cone get quadrupled when the slant height and the radius of the base are doubled as:
 Total surface area = πr (r + l) = π(2r) (2r + 2l) = 4πr (r + l) = 4 × original total surface area.
 Curved surface area = πrl = π(2r)(2l) = 4πrl = 4 × original curved surface area, where "r" is the original radius and "l" is the original slant height of the cone.
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