A right angled triangle has sides of length 3 cm, 4 cm and 5 cm. The triangle is rotated about the side of length 3 cm to generate a right circular cone. What is the volume of the cone formed, in cubic centimetres?

Introduction / Context: This problem connects coordinate geometry of a right angled triangle with the mensuration formula for the volume of a right circular cone. When a right triangle is rotated about one of its sides, the resulting solid is a cone. Identifying which side becomes the height and which side becomes the radius is critical. Once that is clear, we apply the standard cone volume formula. Problems like this check both visualization skills and formula application, and they are common in competitive exams involving quantitative aptitude. Given Data / Assumptions:

• Right triangle sides are 3 cm, 4 cm and 5 cm.
• The triangle is right angled, and 3 cm and 4 cm are the perpendicular sides, 5 cm is the hypotenuse.
• The triangle is rotated about the side of length 3 cm.
• We assume a full rotation of 360 degrees.
• We must find the volume of the resulting cone.

Concept / Approach: In a right triangle with perpendicular sides a and b, and hypotenuse c, if the triangle is rotated about one perpendicular side, say a, then:

• The axis of rotation (side a) becomes the height h of the cone.
• The other perpendicular side b traces a circle and becomes the radius r of the cone base.
• The hypotenuse becomes the slant height, which is not needed for volume.

The volume V of a right circular cone with radius r and height h is: V = (1 / 3) * pi * r^2 * h Step-by-Step Solution: Step 1: Identify height and radius. Triangle sides: 3 cm, 4 cm, 5 cm, with 3 and 4 as perpendicular sides. Rotation is about the side of 3 cm, so height h = 3 cm. Other perpendicular side 4 cm becomes radius r = 4 cm. Step 2: Apply cone volume formula. V = (1 / 3) * pi * r^2 * h V = (1 / 3) * pi * 4^2 * 3 V = (1 / 3) * pi * 16 * 3 V = 16 * pi cubic centimetres. Thus, the volume of the cone formed is 16 pi cubic centimetres. Verification / Alternative check: We can check the choice of height and radius logically. The axis of rotation is fixed, so the distance from any point on the triangle to this axis describes a circle. When rotating around the 3 cm side, the 3 cm side itself stays fixed and forms the axis, so it must be the height. The side opposite the axis, which is 4 cm, moves in a circle at constant distance 4 cm from the axis, so that distance is the radius. This validates our selection of h and r, and hence the computed volume. Why Other Options Are Wrong: Option A (12 pi) and Option B (15 pi) correspond to incorrect combinations or mistaken identification of height or radius. Option D (20 pi) assumes larger dimensions than actually used. Option E (10 pi) is also not consistent with the formula. Only 16 pi cubic centimetres is consistent with the correct substitution of r = 4 cm and h = 3 cm. Common Pitfalls: Many students mistakenly take the hypotenuse as the radius or height. Others mix up which side becomes the axis of rotation. Another common error is to forget the factor 1 over 3 in the cone volume formula. Always clearly identify the rotating side, mark it as the height, and take the perpendicular side as the radius when the triangle is right angled. Final Answer: The volume of the cone formed is 16 pi cubic centimetres.