The amount of three-dimensional space inside a sphere, given by 34πr3.
Imagine filling a sphere with water, then pouring all that water into a cylinder that has the same radius and a height equal to the sphere's diameter (2r). The sphere fills exactly two-thirds of the cylinder. Archimedes was so proud of discovering this relationship that he had it carved on his tombstone.
Showing a random 20 of 50 problems.
Example 1
hard
A sphere is inscribed in a cylinder so that the cylinder's height equals the sphere's diameter. Show the ratio of sphere volume to cylinder volume is 2:3.
Example 2
easy
Use π≈3.14. A sphere has radius 3. Find its volume.
Example 3
medium
A sphere has volume V. If its radius is halved, find the new volume in terms of V.
Example 4
medium
A balloon's volume doubles. By what factor does its radius grow? Give answer to 3 decimal places.
Example 5
easy
A sphere has radius 2. Find its volume (use π≈3.14).Sphere with r = 2 — find volume using π ≈ 3.14
Example 6
hard
A hollow ball has outer radius 5 and inner radius 4. Find the volume of material in terms of π.
Example 7
hard
A solid hemisphere of radius 6 sits on a cylinder of radius 6 and height 4. Find the total volume in terms of π.
Example 8
medium
A spherical ball just fits inside a cube of side 10. Find the volume of the empty space inside the cube, in terms of π.
Example 9
medium
A sphere of radius 5 is dropped into a cylinder of radius 5 partly full of water. By how much does the water level rise, in terms of π then numerically?
Example 10
medium
A balloon's volume increases from 36π to 288π. By what factor did its radius grow?
Example 11
medium
A spherical scoop of ice cream has radius 2 cm. About how many cm³ is it? (Use π≈3.14.)
Example 12
challenge
Earth's radius is about 4 times the Moon's. Approximately how many Moons fit inside the Earth, by volume?
Example 13
easy
What units does the volume of a sphere have if its radius is in meters?
Example 14
easy
A sphere of radius 0 has what volume?
Example 15
hard
If the radius of a sphere is doubled, by what factor does its volume increase? Prove your answer algebraically.
Example 16
challenge
A spherical cap is cut from a sphere of radius 5 by a plane 3 units from the center. Using the cap formula V=3πh2(3R−h), find the smaller cap's volume.
Example 17
easy
A sphere has volume 34π. Find its radius.
Example 18
challenge
Use Cavalieri's principle with a cylinder-minus-two-cones to argue the hemisphere volume is 32πr3.
Example 19
easy
What fraction of a cylinder (radius r, height 2r) does a sphere of radius r fill?
Example 20
medium
A hemisphere of radius 6 sits on top of a cylinder of radius 6 and height 10. Find the total volume in terms of π.