Question

If the radius of base of a cone is 7 cm and its height is 24 cm then find its curved surface area.

Hint:

The triplet (7, 24, 25) is a Pythagorean triplet. Recognizing this allows you to find the slant height of 25 cm instantly without calculating \(\sqrt{625}\), saving valuable time in an exam.

Answer 1

Step 1: Understanding the Concept:
The curved surface area (CSA) of a cone is the area of its lateral surface, excluding the circular base. The formula for CSA involves the radius and the slant height. We must first calculate the slant height using the given radius and perpendicular height.

Step 2: Key Formula or Approach:
1. The slant height (\(l\)) of a cone is related to its radius (\(r\)) and height (\(h\)) by the Pythagorean theorem: \(l = \sqrt{r^2 + h^2}\).
2. The Curved Surface Area (CSA) of a cone is given by the formula: \(CSA = \pi r l\).

Step 3: Detailed Explanation:
We are given:
Radius, \(r = 7\) cm
Height, \(h = 24\) cm
First, we calculate the slant height (\(l\)): \[ l = \sqrt{r^2 + h^2} \] \[ l = \sqrt{7^2 + 24^2} \] \[ l = \sqrt{49 + 576} \] \[ l = \sqrt{625} \] \[ l = 25 \text{ cm} \] Now that we have the slant height, we can calculate the Curved Surface Area using \(\pi = \frac{22}{7}\). \[ CSA = \pi r l \] \[ CSA = \frac{22}{7} \times 7 \times 25 \] The 7 in the numerator and denominator cancels out. \[ CSA = 22 \times 25 \] \[ CSA = 550 \text{ cm}^2 \]

Step 4: Final Answer:
The curved surface area of the cone is 550 cm\(^2\).