Question

A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is:

• A. \(1 : 2 : 3\)
• B. \(2 : 1 : 3\)
• C. \(3 : 1 : 2\)
• D. \(2 : 3 : 1\)

Hint:

When shapes have equal base and height, comparing their volume formulas gives an easy way to find ratios.

Answer 1

The Correct Option is A

Step 1: Use standard volume formulas for each solid.
Let the common base radius be \( r \) and height be \( h \).
Volume of cone = \( \frac{1}{3}\pi r^2 h \)
Volume of hemisphere = \( \frac{2}{3}\pi r^3 \)
But since height of hemisphere is also \( h \), we get \( r = h \)
So, volume = \( \frac{2}{3}\pi h^3 \)
Volume of cylinder = \( \pi r^2 h \)
Step 2: Plug in \( r = h \) to compare volumes.
Cone: \( \frac{1}{3} \pi h^3 \)
Hemisphere: \( \frac{2}{3} \pi h^3 \)
Cylinder: \( \pi h^3 \)
So, ratio of volumes:
\[ \frac{1}{3} : \frac{2}{3} : 1 = 1 : 2 : 3 \]