Volume of Cylinders, Cones & Spheres

A can is a cylinder, an ice cream cone is a cone, a ball is a sphere — all three are what happens when a circle learns to stand up. In Volume we measured boxes by counting cubes; this lesson carries that counting idea into the round world.

Cylinders: base area × height

Slice a cylinder into thin round discs and each disc is a "base". The base area is $\pi r^2$ , stacked $h$ layers high:

$V = \pi r^2 h$

A cylinder of radius 3 and height 5: $V = \pi \times 3^2 \times 5 = 45\pi \approx 141.37$ cm³. At 1000 cm³ per liter that is about 0.14 L — one small glass of milk.

InteractiveFrom Cubes to Cylinders

Length l4Width w3Height h2

V = l × w × h = 4 × 3 × 2 = 24

Bottom layer

Layers

The old cube-counting routine still works here: lay the floor first, then count the layers. A box is "length × width × layers"; a cylinder simply swaps the rectangular floor for a round one — the base area × height idea never moved.

Cones: one third of a cylinder

A cone with the same base and the same height holds exactly one third:

$V = \frac{1}{3}\pi r^2 h$

The water test proves it in seconds: fill the cone, pour it into the matching cylinder, and the third pour fills it to the brim. Pair a cone with that 45π cylinder and you get just $15\pi \approx 47.12$ cm³.

Spheres: only one size matters

A sphere has no length, width or height — its formula needs only $r$ :

$V = \frac{4}{3}\pi r^3$

A sphere of radius 3: $V = \frac{4}{3} \times \pi \times 27 = 36\pi \approx 113.10$ cm³.

Line the three up

The 3 : 2 : 1 coincidence

A cylinder of radius 3 and height 6 (just tall enough to hold that sphere): $V = \pi \times 9 \times 6 = 54\pi$ . The sphere: $36\pi$ . A cone with the same base and height: $18\pi$ . The ratio is 3 : 2 : 1 — one sphere plus one cone fills that cylinder exactly.

How much a container holds is volume in disguise — just swap units (1 cm³ = 1 mL). A cup of radius 5 and height 10 holds about $\pi \times 25 \times 10 = 250\pi \approx 785$ mL, a little under a liter — cups hold more than they look.

Estimate before you compute

Keep two instincts: a cone is about one third of the matching cylinder, and a sphere tucked inside the same-radius cylinder takes about two thirds. Even if a formula escapes you, the order of magnitude will not.

Check yourself

Quick quiz

1. A cylinder of radius 3 and height 5 has what volume?
2. A cone with the same base and height holds what fraction of the cylinder?
3. A sphere of radius 3 has what volume?