Cone volume formula:
why it is exactly one third of a cylinder

Three cones fill one cylinder

Take a cone and a cylinder with the same base radius and the same height. Fill the cone with water and pour it into the cylinder. It takes exactly three cones to fill the cylinder. The same relationship holds for pyramids and prisms, and it is true for every cone whatever its size or proportions.

The proof needs calculus, but the picture is easy to remember. If you know the cylinder volume, a third of it is the cone.

The size of a cone is controlled by two numbers, so quite different shapes can hold the same amount. A cone with r = 3 cm and h = 8 cm has a volume of 1/3 × π × 9 × 8 = 75.40 cm³. A squatter one with r = 4 cm and h = 4.5 cm gives 1/3 × π × 16 × 4.5 = 75.40 cm³ as well. They look nothing alike, yet they would fill the same measuring cup.

Worked example: r = 6 cm, h = 10 cm

The cylinder check is the quickest sanity test. The cone must come out at exactly one third of the cylinder. If it is more, a step went wrong.

The slant height trap

Cones have two lengths that look similar. The height h runs straight up from the middle of the base to the tip. The slant height l runs along the surface from the rim to the tip. They are linked by l = √(r² + h²), so the slant height is always longer.

For the cone above, l = √(36 + 100) = √136 = 11.66 cm. If you used 11.66 as the height, you would get about 439.6 cm³, nearly 17% too high. The slant height is only needed for the curved surface area, π × r × l, which matters if you are covering the outside of a party hat or a funnel.

A pile of sand: using volume to find weight

A heap of dry material forms a rough cone. Measure the width of the base and the height, and the volume follows. To turn it into weight, multiply by the bulk density of the material, which varies a lot with moisture and grading.

Real heaps are not perfectly conical, so treat the result as an estimate within perhaps 10%.

Frustums: cones with the tip cut off

A bucket, a flower pot or a paper cup is a cone with the top removed, called a frustum. Its volume is the large cone minus the small one that was cut off, which simplifies to a single formula.

If r is 0 the shape is a full cone and the formula reduces to π × h × R² ÷ 3, matching the cone formula.

Working backwards to the height or radius

When a funnel, hopper or container is specified by volume, you may need the height. Rearranging gives h = 3V ÷ (π × r²), and for the radius r = √(3V ÷ (π × h)).

Check by going forward: 1/3 × 78.54 × 19.10 = 500.0. The same shape with a 10 cm radius would need only a quarter of the height, 4.8 cm, for the same volume.

Cones, pyramids and the same one-third rule

The factor of one third is not special to round shapes. Any pyramid, whether its base is a square, a triangle or a hexagon, has volume 1/3 × base area × height. A square pyramid with a 6 m by 6 m base and a 10 m height has a volume of 1/3 × 36 × 10 = 120 m³.

A cone is simply the pyramid with a circular base. Once you accept that, a single rule covers roof spires, hoppers, tents, silos with conical bottoms and heaps of sand: take the base area, multiply by the vertical height, and divide by three.

Cones in everyday measurement

Conical shapes show up in kitchens, workshops and on roads. A drinks funnel, a piping bag, a traffic cone and the lower part of a grain hopper are all cones or frustums. The volume is rarely the whole story: a hopper's slope also decides whether the contents flow, which depends on friction between the material and the wall.

When a calculation is for something you will buy by volume, convert at the end: 1 cm³ is 1 mL and 1 m³ is 1,000 litres. When it is for something you will buy by weight, the density is the missing piece, and it should come from a supplier rather than from a general table.

Scaling and checks

• Ice-cream cone of radius 3 cm and height 12 cm holds about 113 cm³ before the scoop.
• A funnel has the same maths, but check the neck is not a separate cylinder.
• Volume in cm³ is millilitres for liquids.

A cone volume calculator is worth using when you have only the slant length and radius, or want to compare several heights quickly.