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).Find the volume using π ≈ 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 r and total height 2h (each cone has height h). What fraction of the cylinder's volume is sand-fillable (i.e., total cone volume)?
Example 5
medium
A cone has volume 100π and height 12. Find its radius.Volume = 100π; find the radius.
Example 6
easy
A cone and a cylinder have the same radius and height. The cylinder's volume is 90π cm³. What is the cone's volume?
Example 7
medium
A cone has a base radius 6 and height 9. A cylinder is inscribed with the same height and a radius such that the cylinder's top fits exactly inside the cone at height h... 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 3 and height 4. Find its volume in terms of π.Find the volume in terms of π.
Example 10
medium
A cone has volume 75π and height 9. Find its radius.Volume = 75π; find the radius.
Example 11
easy
A cone has radius 4 and height 6. Find its volume in terms of π.Find the volume in terms of π.
Example 12
easy
Fill the blank: cone volume uses the ___ height (perpendicular), not the slant height.
Example 13
medium
Two cones have radii 3 and 6 and the same height. Find the ratio of their volumes (small : large).
Example 14
medium
A cone of radius 6 and height 12 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 75π cm³ and a radius of 5 cm. Find the height of the cone.Volume = 75π cm³; find the height.
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 R (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.Find the volume in terms of π.
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.
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.