Pyramids and Cones

Geometry · Axiom Academy

Pointed shapes that taper to a vertex — and why their volume is exactly one-third of the prism or cylinder around them. Take a cone and the cylinder with the same base and the same height . How many cone-fulls does it take to fill the cylinder? Exactly three . The same is true for a pyramid and its matching prism — and that single fact is the whole reason for the one-third in every formula below. Why exactly a third? Watch a cube split into three identical pyramids — same shape, same size — that fold together to fill it with nothing left over. Each one is therefore exactly one-third of the cube around it. B = area of the base · h = perpendicular height Volume works the same way for both — watch a pyramid round off into a cone. The rule V = ⅓Bh never changes; only the base area B switches from a square's s² to a circle's πr² . The surface area differs because a pyramid has flat triangular faces while a cone has one rolled-up curved face. Two different "heights" appear in these formulas, and mixing them up is the classic mistake. Watch how the height, the radius, and the slant height form a right triangle inside the cone. The perpendicular distance from the base straight up to the apex. This is the one used for volume . The distance along the surface from a base edge up to the apex. This is the one used for surface area . A square pyramid has a base with side 6 m and height 8 m. Find its volume. A cone has radius 5 cm and height 12 cm. Find its volume. (Use π ≈ 3.14)

This is the written version of the interactive lesson above. See the full Geometry course.