Practice Volume of a Cone in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The amount of three-dimensional space inside a cone, which is exactly one-third the volume of a cylinder with the same base and height.

Imagine filling a cone-shaped paper cup with water and pouring it into a cylinder of the same width and height. You'd need to fill the cone exactly three times to fill the cylinder. A cone is a cylinder that 'tapers to a point,' losing two-thirds of its volume in the process.

Showing a random 20 of 50 problems.

Example 1

easy
Why does the cone volume have a factor of 1/3?

Example 2

easy
A cone has radius 6 and height 5. Find its volume (use π≈3.14\pi \approx 3.14).

Example 3

medium
A cone is filled with sand and emptied into a cylinder of the same base and height. The cylinder ends up filled to what fraction?

Example 4

challenge
A sand timer is two cones joined point-to-point inside a cylinder of radius rr and total height 2h2h (each cone has height hh). What fraction of the cylinder's volume is sand-fillable (i.e., total cone volume)?

Example 5

medium
A cone has volume 100π100\pi and height 12. Find its radius.

Example 6

easy
A cone and a cylinder have the same radius and height. The cylinder's volume is 90π90\pi cm³. What is the cone's volume?

Example 7

medium
A cone has a base radius 66 and height 99. A cylinder is inscribed with the same height and a radius such that the cylinder's top fits exactly inside the cone at height hh... actually find the ratio of cone volume to a cylinder of the same base and height.

Example 8

easy
What is the formula for the volume of a cone?

Example 9

easy
A cone has radius 33 and height 44. Find its volume in terms of π\pi.

Example 10

medium
A cone has volume 75π75\pi and height 99. Find its radius.

Example 11

easy
A cone has radius 44 and height 66. Find its volume in terms of π\pi.

Example 12

easy
Fill the blank: cone volume uses the ___ height (perpendicular), not the slant height.

Example 13

medium
Two cones have radii 33 and 66 and the same height. Find the ratio of their volumes (small : large).

Example 14

medium
A cone of radius 66 and height 1212 has its top third (by height) cut off by a plane parallel to the base. Find the volume of the small cone that was removed.

Example 15

medium
An ice cream cone has a volume of 75π75\pi cm³ and a radius of 5 cm. Find the height of the cone.

Example 16

challenge
A frustum has radii 3 (top) and 6 (bottom) and height 4. Find its volume by extending to the full cone.

Example 17

challenge
Among all cones inscribed in a sphere of radius RR (with vertex at the top of the sphere and base parallel to a horizontal plane through the sphere), find the height of the cone with maximum volume.

Example 18

easy
A cone has a radius of 6 cm and a height of 9 cm. Find its volume. Leave your answer in terms of π\pi.

Example 19

challenge
An inverted conical tank (point down) has radius 5 at the top and height 10. Water fills it to depth 6. Find the water volume in terms of π\pi.

Example 20

challenge
Derive the cone volume formula using Cavalieri's principle and the fact that a cube can be split into 3 equal pyramids.
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