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The cone is a 3 Dimensional shape. Similar to other 3D shapes, a cone has both flat and curved surfaces. The curved surface area of a cone specifically refers to the rounded part of the cone, excluding the flat circular base. To calculate the curved surface area (CSA) of a cone, we multiply the radius and slant height by pi (π). This article will discuss details about the Curved Surface Area Of the Cone.
Curved Surface Area Of Cone Class 9
The surface area of a cone is the area covered by its outer surface. A cone is a 3-D shape with a circular base, having a radius and diameter. The distance from the base’s centre to the cone’s top is its height. Surface area usually means the total area, combining the curved and flat circular areas.
To calculate the surface area of a cone, we consider both the curved and flat circular areas. The curved surface area, however, only involves the curved part of the cone.
Curved Surface Area Of Cone Formula
A cone has two types of surface areas:
Total Surface Area (TSA) of Cone:
The total surface area of a cone (TSA) is calculated using the formula
TSA = πr(r + l),
where ‘r’ is the radius and ‘l’ is the slant height.
Curved Surface Area (CSA) of Cone:
The curved surface area of a cone (CSA) is calculated using the formula
CSA = πrl,
where ‘r’ is the radius, and ‘l’ is the slant height.
These formulas can also be expressed in terms of the cone’s height (‘h’). The height is different from the slant height, and they are related by the Pythagorean theorem:
h² + r² = l²
Solving for ‘l’, we get,
l = √(h² + r²).
Therefore, in terms of height, the formulas become:
- Total Surface Area (TSA):
TSA = πr(r + l) can be expressed as TSA = πr[r + √(h² + r²)].
- Curved Surface Area (CSA):
CSA = πrl can be expressed as CSA = πr[√(h² + r²)].
Curved Surface Area Of Cone: Derivation
To calculate the curved surface area (CSA) of a cone, we multiply the radius and slant height by pi (π). Let’s derive the formula for the curved surface area of a cone, also known as the CSA.
The formula for the Curved Surface Area (CSA) of a cone is given by:
CSA of cone = πrl
Where:
r is the radius, l is the slant height, and π is a mathematical constant approximately equal to 3.14159.
The curved surface area is also referred to as the lateral surface area of a cone. This formula helps us find the amount of curved surface that covers the cone, excluding the circular base.
Lateral Surface Area Of Cone
The curved surface area refers only to the lateral surface. Therefore, The curved surface area of a cone (CSA) is calculated using the formula
CSA = πrl,
where ‘r’ is the radius, and ‘l’ is the slant height.
Curved Surface Area Of Right Circular Cone
A right circular cone is a special type of cone characterised by its axis, which is the line connecting the vertex (top point) to the midpoint of the circular base. This line forms a right angle with the circular base. In simpler terms, the centre of the circular base is connected to the tip of the cone, creating a right-angled triangle.
Below given are a few formulas related to the right circular cone.
Curved Surface Area (CSA) of a Right Circular Cone:
CSA = πrl
Total Surface Area (TSA) of a Right Circular Cone:
TSA = π(r + l)r
Volume of a Right Circular Cone:
V = ⅓ πr2h
Curved Surface Area Of Frustum Of Cone
The formula for the Curved Surface Area of frustum of a cone is given by:
CSA = πl(R + r)
Where:
r is the radius of the smaller circle (top circular base of the frustum),
R is the radius of the larger circle (bottom circular base of the frustum),
l is the slant height of the frustum, and
π is the mathematical constant pi, approximately equal to 3.14159.
Curved Surface Area: Example
Example: Find the total surface area and curved surface area of the cone whose radius is 5 cm and slant height is 8 cm.
Solution: Given:
Radius r = 5 cm
Slant Height l = 8 cm
Curved Surface Area (CSA):
CSA = πrl
CSA = 3.14*(5)*8 = 125.6 square cm
Total Surface Area (TSA):
TSA = πr(r + l)
TSA = 3.14*5*(5 + 8) = 196.7 square cm
Curved Surface Area: Practice Questions
Example 1: Determine the total surface area and curved surface area of a cone with a radius of 6.5 meters and a slant height of 10 meters.
Example 2: Calculate the total surface area and curved surface area of a cone having a radius of 9 inches and a slant height of 12 inches.
Example 3: Find the total surface area and curved surface area of a cone with a radius of 3.2 cm and a slant height of 6.4 cm.
Example 4: Determine the total surface area and curved surface area of the cone when the radius is 12.5 cm and the slant height is 15 cm.
FAQs on Curved Surface Area of Cone
What is the curved surface area (CSA) of a cone?
The curved surface area (CSA) of a cone refers to the measurement of the total surface area of its curved or lateral surface, excluding the flat circular base.
How is the curved surface area (CSA) of a cone calculated?
The formula for the curved surface area of a cone is given by CSA = πrl, where r is the radius of the base, and l is the slant height.
What is the difference between the curved surface area (CSA) and the total surface area (TSA) of a cone?
The curved surface area (CSA) involves only the curved part of the cone, while the total surface area (TSA) includes both the curved and flat circular areas.
