The slant height of a right circular cone is 10m and its height is 8m. Find the area of its curved surface?

AIIMS Raipur - 2017 (shift-2)

Reasoning

Hard

Appeared in: AIIMS Raipur - 2017 (shift-2)

Explanation

The curved surface area (CSA) of a cone is found using the formula CSA = πrl, where 'r' is the radius and 'l' is the slant height.
First, the radius 'r' must be calculated using the Pythagorean theorem, as the cone's height (h), radius (r), and slant height (l) form a right-angled triangle: l² = h² + r².
Given l = 10 m and h = 8 m, the radius is calculated as r = √(10² - 8²) = √(100 - 64) = √36 = 6 m.
Substituting the radius (r=6 m) and slant height (l=10 m) into the CSA formula gives: CSA = π × 6 × 10 = 60π m².

Option A: This value is exactly half of the correct answer. It may result from an arithmetic error, such as incorrectly dividing the final product by 2.
Option C: This value is obtained by incorrectly multiplying π by the height and the slant height (π × 8 m × 10 m = 80π m²). The correct formula uses the radius, not the height.
Option D: This value may result from incorrectly using the height in place of the slant height and making another error, for example, π × r × h = π × 6 × 8 = 48π m², which is close but still incorrect. 40π m² does not follow a clear logical error.

Visual 1: Diagram - A diagram of a right circular cone illustrating the vertical height (h), the radius of the base (r), and the slant height (l). The diagram should highlight that these three lengths form a right-angled triangle, visually representing the relationship l² = h² + r².

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Calculation of the curved surface area of a right circular cone as background academic context rather than a clinical decision trigger.
This question tests fundamental knowledge of geometry, which is a component of the numerical ability section in nursing entrance exams.
Strong foundational math skills are essential for nurses for tasks such as medication dosage calculations, IV drip rate regulation, and interpreting patient data charts.
What if the question asked for the Total Surface Area? You would first calculate the curved surface area (60π m²) and then add the area of the circular base (πr² = π × 6² = 36π m²). The total surface area would be 60π + 36π = 96π m².

How to Approach the Question

First, identify all the given information in the problem: slant height (l) = 10 m and height (h) = 8 m.
Next, determine what the question is asking for: the area of the curved surface (CSA).
Recall the formula for the curved surface area of a cone: CSA = πrl. You will notice that the radius (r) is not given directly.
Recognize that the height, radius, and slant height of a right circular cone form a right-angled triangle. Apply the Pythagorean theorem (l² = h² + r²) to find the missing radius.
Substitute the given values into the theorem (10² = 8² + r²) and solve for 'r'.
Once the radius is found, substitute the values of 'r' and 'l' into the CSA formula to calculate the final answer.

Concept Tested & Keywords

Concept Tested: Calculation of the curved surface area of a right circular cone.
Stem keywords: right circular cone, slant height, height, curved surface area
Lead-in keywords: Find

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