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Surface Area of Cone

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The surface area of a cone is the total area encompassing the circular base as well as the curved surface of the cone. A cone has two types of surface areas. If the radius of the base is ‘r’ and the slant height is ‘l’, we use two formulas:

  1. Total Surface Area (TSA) of the cone = πr(r + l)
  2. Curved Surface Area (CSA) of the cone = πrl.

Let’s learn about the surface area of the cone, including the formulas for TSA and CSA, with the help of solved examples.

Surface Area of Cone

Examples of Cone

What is the Surface Area of a Cone?

The surface area of a cone is visualized as the area occupied by the cone when it is cut open. It is formed by a circular base and a curved surface. The surface area of the cone is dependent on the radius of its base and the height of the cone. There are two types of surface areas for cones.

Surface-Area-of-Cone

Surface Area TypeFormulaUnits
Curved Surface Area (S)​ πr√(r2 + h2)square units
Total Surface Area (T)πr2 + πr√(r2 + h2)square units

Cone Surface Area Definition

Cone is a three-dimensional-shaped geometric figure that has a flat face and a curved surface with a pointed end. TheA cone Area of a conesurface area is the total area occupied by the surfaces of the cone. There are two types of surface areas of Cone: curvedcones Surface areassurface and totalareas Surface areassurface of cones.

Surface Area of Cone Formula

The surface area of a cone is defined as the area occupied by the boundary or the surface of the cone. A cone has two kinds of surface areas, namely, a curved surface area and a total surface area.

Surface Area of Cone Formula

Surface Area of Cone Formula

Curved Surface Area of Cone

The curved surface of a cone is defined as the area of the curved part of the cone, i.e., the area of the cone excluding its base. It is also known as the lateral surface area of the cone.

The formula for the CSA (curved surface area) of the cone is given as follows:

Curved Surface Area of Cone

Curved Surface Area of Cone

where,

  • r” is the radius of the base of a cone
  • l” is the slant height of the cone

Total Surface Area of Cone

The total surface area of a cone is defined as the total area occupied by a cone in a three-dimensional space, i.e., the area of the curved surface and the area of the circular base. The formula for the TSA (total surface area) of the cone is given as follows:

Total Surface Area of Cone

Total Surface Area of Cone

where,

  • r” is the radius of the base of a cone
  • l” is the slant height of the cone

Derivation of Surface Area of Cone Formula

To observe the figure formed by the surface of a cone, take a paper cone and then cut it along its slant height. Now, mark A and B as the two endpoints and O as the point of the intersection of the two lines. Now if we open this, it will look like a sector of a circle.

Surface Area of Cone Derivation

So, to find the curved surface area of the cone, we have to find the area of the sector.

Area of sector in terms of length of arc = (arc length × radius)/2 = ((2πr) × l)/2 = πrl.

CSA of a Cone  = πrl square units

Total surface area of a cone (T) =  Area of the base + Curved Surface area

Since the base is a circle, the area of the base is πr2

⇒ T = πr2 + πrl  = πr(r + l)

TSA of Cone = πr (r + l) square units

Learn More:

Surface Area of Cone and Height

Considering the slant height, height, and radius of the cone, they form a right-angle triangle, where slant height is the hypotenuse, the base is the radius of the base, and height is the altitude of the right-angle triangle.

Surface Area of Cone and Height Relation

Using Pythagoras Theorem,  we get l2 = r2 + h2.2+h2

Thus, slant height of a cone (l) = √(r2 + h2)

So, by replacing the value of slant in the surface areas formula of a cone, we get

Curved surface area (S) = πr√(r2 + h2) square units

Total surface area (T) = πr2 + πr√(r2 + h2) square units

Surface Area of Cone Examples

Example 1: Find the total surface area of a cone if its radius is 15 cm and its slant height is 10 cm. (Use π = 3.14.formula)

Solution: 

Given

  • Radius of cone (r) = 15 cm
  • Slant height (l) = 10 cm

We know that,

Total surface area of the cone = πr (r + l) square units

= (3.14) × 15 × (15 + 10)

= 1,177.5‬ sq. cm

Hence, the total surface area of the cone is 1,177.5‬ sq. cm.

Example 2: What is the height of a cone if its radius is 14 units and its curved surface area is 1100 square units? (Use π = 22/7)

Solution: 

Given

  • Radius of cone (r) = 14 units
  • Curved surface area of the cone = 1100 square units 

Let the slant height of the cone be “l” and the height of the cone be “h”.

We know that,

Curved surface area of the cone = πrl square units 

⇒ 1100 = (22/7) × 14 × l

⇒ 44 × l = 1100

⇒ l = 1100/44 = 25 units

We know that,

slant height (l) = √(h2 + r2)

⇒ h = √(l2 – r2) 

= √(252 – 142) = √429 = 20.71 units

Thus, the height of the cone is 20.71 units.

Example 3: Determine the slant height of the cone if the total surface area of the cone is 525 sq. cm and the radius is 7 cm. (Use π = 22/7)

Solution: 

Given

  • Radius of cone (r) = 7 cm
  • Total surface area of the cone = 525 sq. cm

Let the slant height of the cone be “l”.

We know that,

The total surface area of the cone = πr (r + l) square units

⇒ (22/7) × 7 × (7 + l) = 525

⇒ 22 × (7 + l) = 525

⇒ 7 + l = 25

⇒ l = 18 cm

Therefore, the slant height of the cone is 18 cm.

Surface Area of Cone Class 9 NCERT

Find Solutions to Exercise of Class 9 NCERT Chapter 13 Surface Area and Volumes to practice and hone your knowledge and understanding of the concept.

Surface Area of Cone Class 9 Extra Questions

Surface area of cone class 9 worksheet and High Order Thinking Skills (HOTS) questions are provided below:

Q1. A right circular cone has a radius of 5 cm and a slant height of 12 cm. Calculate its total surface area.

Q2. The curved surface area of a cone is 1003.14. square centimeters. If its radius is 6 cm, find its slant height.

Q3. A cone has a total surface area of 200100 square centimeters. If its slant height is 10 cm, find its radius.

Q4. The radius of a cone is tripled while its slant height remains constant. How does its total surface area change?

Q5. Two cones have the same curved surface area. If one cone has a radius twice that of the other, compare their heights.

Practice Questions on Cone Surface Area

Q1. Find the CSA and TSA of the200 cone if it radius and height areits 5 cm and 12 cm respectivelycm,.

Q2. If slant height is 12 cm and theIf the base radius is 7 cm, find curved surface area and thefind the total surface area of the cone.

Q3. Find Total totalthe Area of Cone if CSA is 144 cm2 and base radius is 7 cm.

Q4. Find Curved curvedthe Area of Cone if Radius is 14 cm and Slant slant heightits is 20 cm.

Related:

FAQs on Surface Area of Cone

What happens to curved surface area of ato the cone when its height is doubled?

Curved Surface area of the cone directly depends on the radius of its base.

Curved surface area = πrl 

If the radius of cone is doubled its curved surface area also gets doubled.

How do Iof a find surface area of afind the cone?

Surface area of cone can be calculated in two ways,

CSA = πrl

TSA = πrl(r+l)

How do Iof a calculate slant height of acalculate the cone?

Slant height of a cone is defined by the formula

l=√(r2 + h2) units

Write the formula for base surface of afor the cone.

Surface of the base of a Cone is circular and the formula for the base of surface of the cone is πr2 square units.

What is meant by surface area of cone?

Surface Area of a Cone is the region occupied by the Surface of a Cone in 3D space. It can be calculated by finding the sum of the lateral area and the base area of the cone.

What is the net surface area of a cone?

Net surface area of a cone refers to the total surface area of the cone, including both the curved surface area and the area of the base. It represents the entire external surface of the cone when all parts are unfolded and laid flat.



Last Updated : 21 Feb, 2024
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