The curved surface area of a cone is calculated using the formula πrl, where r is the radius and l is the slant height. For a cone with a 10.5 cm diameter (5.25 cm radius) and a 10 cm slant height, the area is 52.5π cm². Using π ≈ 3.14, it approximates to 164.93 cm².

Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its surface area.

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Understanding the Curved Surface Area of a Cone

The concept of curved surface area is particularly important in geometry, especially when dealing with three-dimensional shapes like cones. The curved surface area refers to the area of the outer surface of the cone, excluding its base. This concept is widely used in various fields, including architecture, engineering, and design.

Basic Formula for Curved Surface Area

The formula to calculate the curved surface area of a cone is given by πrl, where r is the radius of the base of the cone, and l is the slant height of the cone. This formula is derived from the properties of a circle and the Pythagorean Theorem, considering that a cone is essentially a circle rotated around one of its diameters.

Given Dimensions of the Cone

In our specific problem, we are given the diameter of the base of the cone as 10.5 cm and the slant height as 10 cm. The diameter is the total distance across the circle that forms the base of the cone. The slant height is the distance from the top of the cone (vertex) to any point on the rim of the base, measured along the surface of the cone.

Calculating the Radius

The first step in our calculation is to determine the radius of the base of the cone. The radius is half of the diameter. Therefore, for a cone with a diameter of 10.5 cm, the radius r is calculated as r = 10.5/2 = 5.25 cm.

Applying the Formula

With the radius and the slant height known, we can apply the formula for the curved surface area. Substituting the values into the formula πrl, we get π × 5.25 × 10.
This calculation will give us the curved surface area of the cone in square centimeters.

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Final Calculation and Result
To get a numerical value, we use the approximation π ≈ 3.14. Thus, the curved surface area is approximately 52.5 × 3.14 ≈ 164.93 cm². Therefore, the curved surface area of the cone with the given dimensions is approximately 164.93 cm². This calculation is crucial for practical applications where precise measurements are needed, such as in construction and manufacturing.

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Questions of 9th Maths Exercise 11.1 in Detail

Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area.
Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
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What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base radius 6 m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20 cm (Use π = 3.14).
The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of ₹210 per 100 m².
A joker’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.
A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is ₹12 per m², what will be the cost of painting all these cones?

Last Edited: January 2, 2024