A cone is a 3-dimensional geometric figure with a circular base and a pointed top called the apex. The distance between the center of the circular base and the apex is the height of the cone. The surface of the cone curves smoothly from the edge of the base to the apex. It is a common shape found in objects like ice cream cones, traffic cones, and funnels.
A cone has two types of surface areas.
- Total Surface Area (TSA) of the cone
- Curved Surface Area (CSA) of the cone
Here, we will learn about the surface area of the cone, including the formulas for Total Surface Area and Curved Surface Area, with the help of solved examples.
Given below is the illustration of a Cone with the formula to calculate the Surface area:
Surface Area of ConeAs discussed earlier 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 and their formulas are given below.
Total Surface Area of Cone
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 a 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 = πr(r + l) Square Units
where,
- "r" is the Radius of the Base of a Cone
- "l" is the Slant Height of the Cone
Curved Surface Area of Cone
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 (S) = πrl Square Units
where,
- "r" is the Radius of the Base of a Cone
- "l" is the Slant Height of the Cone
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 intersection of the two lines. Now, if we open this, it will look like a sector of a circle.
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 a Cone with Slant Height
Considering the slant height, height, and radius of the cone, they form a right-angle triangle, where the slant height is the hypotenuse, the base is the radius of the base, and the height is the altitude of the right-angle triangle.
Using Pythagoras' Theorem, we get l2 = r2 + h2.2+h2
Thus, the slant height of a cone (l) = √(r2 + h2)
So, by replacing the value of slant in the surface area formula of a cone, we get
Curved Surface Area (CSA) = πr√(r2 + h2) square units
Total Surface Area (TSA) = πr2 + πr√(r2 + h2) square units
Volume of a Cone
The volume of a cone is the measure of the amount of space enclosed within the cone. It tells us how much material or substance the cone can hold. Mathematically, it is one-third the volume of a cylinder with the same base and height.
Volume Formula:
Volume = 1/3 (πr2h)
Solved Question on Surface Area of Cone
Question 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 Cone = πr (r + l) square units
= (3.14) × 15 × (15 + 10)
= 1,177.5 sq. cm
Hence, total surface area of cone is 1,177.5 sq. cm.
Question 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, height of cone is 20.71 units.
Question 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, slant height of cone be "l"
We know that,
Total surface area of Cone = πr (r + l) square units
⇒ (22/7) × 7 × (7 + l) = 525
⇒ 22 × (7 + l) = 525
⇒ 7 + l = 23.86
⇒ l = 16.86cm
Therefore, slant height of cone is 16.86 cm.
Question 4: The radius of a cone is 9 cm, and its curved surface area is 407 cm². Calculate the height of the cone.
Solution:
Given
- Radius of cone (r) = 9 cm
- Total surface area of the cone = 407 cm2
Let, slant height of cone be "l"
We know that,
CSA = πrl square units
⇒ (22/7) × 9 × l = 407
⇒ 22 × 198/7 × l = 407
⇒ l = 2849/198
⇒ l = 14.39cm
Use Pythagoras' theorem to find height h
l2 = r2 + h2
(14.39)2 = 92 + h2
207.09 - 81 = h2
h = √126.09
h = 11.23 cm
Therefore, height of cone is 11.23 c
Surface Area of Cone Class 9 NCERT Solutions and Worksheet
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 worksheet and High Order Thinking Skills (HOTS) questions are provided below:
Question 1. A right circular cone has a radius of 5 cm and a slant height of 12 cm. Calculate its total surface area.
Question 2. The curved surface area of a cone is 1003.14. square centimetres. If its radius is 6 cm, find its slant height.
Question 3. A cone has a total surface area of 200100 square centimetres. If its slant height is 10 cm, find its radius.
Question 4. The radius of a cone is tripled while its slant height remains constant. How does its total surface area change?
Question 5. 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
Question 1. Find the CSA and TSA of the cone if its radius and height are 5 cm and 12 cm, respectively.
Question 2. If the slant height is 12 cm and the base radius is 7 cm, find the curved surface area and total surface area of the cone.
Question 3. Find the total surface area of the cone if the CSA is 144 cm2 and the base radius is 7 cm.
Question 4. Find the curved surface area of the cone if the radius is 14 cm and Slant slant height is 20 cm.
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Total Surface Area
Lateral or Curved Surface Area