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How Volume Is Calculated

Five shapes, five formulas: cube = edge³, rectangular box = a×b×c, cylinder = π × r² × height, sphere = (4/3) × π × r³, and cone = (1/3) × π × r² × height. Notice that a cone with the same base radius and height as a cylinder holds exactly one-third of that cylinder's volume — not roughly a third, but exactly, a result provable through integral calculus (integrating the area of circular cross-sections that shrink linearly to zero at the apex). Pour water from a cone-shaped container into a cylinder sharing its base and height, and you'd need to do it three times to fill the cylinder.

Concrete example: a cone with radius 3 cm and height 10 cm has volume (1/3) × π × 3² × 10 = (1/3) × π × 90 ≈ 94.2 cm³. A cylinder with the same 3 cm radius and 10 cm height has volume π × 3² × 10 ≈ 282.7 cm³ — almost exactly three times the cone's volume, confirming the 1/3 relationship down to rounding.

What's Worth Knowing

• The cone-cylinder 1/3 rule: holds for any radius and height, as long as both shapes share the same base and the same height — it's a fixed geometric ratio, not an approximation.
• Units cube automatically: entering cm returns cm³ (1,000 cm³ = 1 litre); entering metres returns m³ (1 m³ = 1,000 litres).
• Partially filled containers: for a box or an upright cylinder, use the liquid's own height instead of the container's full height to get the filled volume directly.
• Sphere vs cylinder: a sphere inscribed exactly inside a cylinder (same radius, height = diameter) fills two-thirds of that cylinder's volume — a lesser-known cousin of the cone rule, also provable with the same integration technique.
• Doubling a dimension doesn't double the volume: because radius and edge length are raised to the second or third power, doubling a cube's edge multiplies its volume by 8, not 2 — a detail that trips up quick mental estimates when comparing container sizes.