Cone, Pyramid & Sphere Volumes

Anything that tapers to a point holds one third. A cone is a third of its cylinder and a pyramid is a third of its prism, and it is the same one third in both cases: the rule is base area times height divided by three, whatever the base shape happens to be.

A sphere fills exactly two thirds of the cylinder that just contains it. Archimedes proved this and had it carved on his tomb. For a sphere the height is 2r, so the bounding cylinder is 2πr³, and two thirds of that is the familiar ⅔πr³.

Slant height is not vertical height. A cone's curved surface uses the slant l = √(r² + h²), not h. Using h by mistake is one of the most common errors in Module 1 volume questions, and the answer comes out too small.

• Try this: With the cone showing, read the share of the bounding cylinder. Now change r and h. Does the share ever move?
• Try this: Switch to the sphere. What share does it take, and what happens to the height slider?
• Try this: For a cone with r = 3 and h = 4, work out the slant height in your head, then check it against the readout.