Question

Consider a square sheet of side 1 unit. In the first step, it is cut along the main diagonal to get two triangles. In the next step, one of the cut triangles is revolved about its short edge to form a solid cone. The volume of the resulting cone, in cubic units, is _________

• A. \( \frac{\pi}{3} \)
• B. \( \frac{2\pi}{3} \)
• C. \( \frac{3\pi}{2} \)
• D. \( 3\pi \)

Hint:

To find the volume of a cone formed by revolving a triangle, use the formula \( V = \frac{1}{3} \pi r^2 h \), where \( r \) is the radius and \( h \) is the height of the cone.

Answer 1

The Correct Option is A

We are given a square sheet with side 1 unit, and the triangle is formed by cutting along the diagonal. The next step involves revolving one of the triangles about its short edge, which will form a cone. Let's find the volume of this cone.
- The base radius \( r \) of the cone is half of the side of the square, so \( r = \frac{1}{2} \).
- The height \( h \) of the cone is the length of the other side of the triangle, which is also \( 1 \).
The formula for the volume of a cone is: \[ V = \frac{1}{3} \pi r^2 h. \] Substituting the values of \( r \) and \( h \): \[ V = \frac{1}{3} \pi \left( \frac{1}{2} \right)^2 \times 1 = \frac{1}{3} \pi \times \frac{1}{4} = \frac{\pi}{3}. \] Thus, the volume of the cone is \( \frac{\pi}{3} \) cubic units. Final Answer: \( \frac{\pi}{3} \)