Cone and pyramid volumes

A cone holds exactly one third of a cylinder with the same base and height: V = (1/3)πr²h.

Check it at home. Fill a paper cone with salt and pour it into a tin with the same base and height: it takes three cones to fill the tin. Or shape three clay cones from one clay cylinder.

Archimedes (c. 225 BCE) first found this formula. "Why exactly 1/3?" A full proof needs calculus, in Grade 12.

Example. Turning a 5-12-13 triangle about the side 12 makes a cone with h = 12 and r = 5: V = (1/3)π × 25 × 12 = 100π cm³.

A surprise. Water to half the depth of a conical cup fills a small cone with half the height and half the radius: (1/2)³ = 1/8 of the cup, not half.

Pyramids work the same way: V = (1/3) × area of base × height, whatever shape the base is. A cone is just a pyramid on a circular base.