Question

# A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio $$1:2:3$$.

Solution

## Let the radius of cone, hemisphere and cylinder be r. Then, height of the hemisphere = r, height of the cone = r, and height of the cylinder = r. Let $$V_{1},V_{2},V_{3}$$ be the volumes of cone, hemisphere and the cylinder respectively. Then, $$V_{1} = \dfrac{1}{3}\pi r^{2}\times r = \dfrac{1}{3}\pi r^{3}$$$$V_{2} = \dfrac{2}{3}\pi r^{3}$$ $$V_{3} = \pi r^{2}\times r = \pi r^{3}$$$$\therefore V_{1}:V_{2}:V_{3} = 1:2:3$$Mathematics

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