A right circular cone has slant height 10 m and vertical height 8 m. Find the curved surface area of the cone in square metres.

Introduction / Context:
This question deals with the geometry of a right circular cone. You are given the slant height and the vertical height, and you are asked to find the curved surface area. The curved surface area formula uses the radius and slant height, so we must first determine the radius using Pythagoras theorem applied to the right triangle formed by the radius, height and slant height. This is a standard problem that checks both geometric understanding and formula usage.

Given Data / Assumptions:

• Slant height of the cone, l = 10 m.
• Vertical (perpendicular) height of the cone, h = 8 m.
• The cone is a right circular cone.
• We must find curved surface area (CSA) in square metres.

Concept / Approach:
For a right circular cone, we use two main formulas:
l^2 = r^2 + h^2 (from Pythagoras theorem) Curved surface area = pi * r * l First, we compute the radius r from the relation between slant height, height and radius. Once r is known, we substitute into the curved surface area formula and simplify to get the final expression in terms of pi.

Step-by-Step Solution:
Given l = 10 m, h = 8 m. From Pythagoras theorem: l^2 = r^2 + h^2. 10^2 = r^2 + 8^2, so 100 = r^2 + 64. Therefore r^2 = 100 − 64 = 36. Thus r = sqrt(36) = 6 m. Curved surface area = pi * r * l = pi * 6 * 10. Curved surface area = 60 pi square metres.

Verification / Alternative check:
We can verify by approximate numeric value. Using pi ≈ 3.14, CSA ≈ 60 * 3.14 ≈ 188.4 m^2. A cone with radius 6 m and slant height 10 m should indeed have an area of around 190 m^2, which feels reasonable for these dimensions. None of the other options, when multiplied by pi, matches the correct product of 6 and 10.

Why Other Options Are Wrong:
40 pi, 50 pi, 70 pi and 80 pi sq.m correspond to either using incorrect values of radius or slant height, or miscalculating the Pythagoras relation. For example, 40 pi could arise from mistakenly using r = 4 m, and 80 pi might come from incorrectly multiplying 8 and 10 without first verifying the radius. Only 60 pi comes from the correct combination of r = 6 m and l = 10 m.

Common Pitfalls:
Students sometimes confuse height and slant height, using the height directly in the CSA formula instead of the slant height. Another error is to forget to take the square root when computing the radius from r^2. It is also common to mistakenly use the total surface area formula, which includes the base area, instead of the curved surface area formula required here.

Final Answer:
The curved surface area of the cone is 60 pi sq.m.