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 ฯ€โ‰ˆ3.14\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 100ฯ€100\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 90ฯ€90\pi cmยณ. 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 ฯ€\pi.

Example 10

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

Example 11

easy
A cone has radius 44 and height 66. Find its volume in terms of ฯ€\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 75ฯ€75\pi cmยณ 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 ฯ€\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 ฯ€\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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