Volume Formula Chart
| Shape | Volume | Surface area |
|---|---|---|
| Rectangular box | l × w × h | 2(lw + lh + wh) |
| Cube | a³ | 6a² |
| Cylinder | πr²h | 2πr(r + h) |
| Sphere | 4/3 πr³ | 4πr² |
| Cone | ⅓ πr²h | πr(r + slant) |
| Pyramid | ⅓ × base area × h | base + lateral faces |
| Capsule | πr²h + 4/3 πr³ | 2πr(2r + h) |
The One-Third Rule
A cone holds exactly one third of the cylinder with the same base and height. A pyramid holds one third of the prism with the same base and height. This is not a coincidence — it follows from integrating the cross-sectional area, and it holds for any shape that tapers linearly to a point.
Archimedes proved that a sphere occupies exactly two thirds of its circumscribing cylinder, and considered it his finest result. He had the figure carved on his tomb.
Volume Scales With the Cube
Doubling every dimension multiplies volume by eight while surface area only quadruples. This square-cube law explains a great deal of the physical world: why large animals need proportionally thicker legs, why small animals lose heat faster, and why crushed ice cools a drink faster than a single block of the same mass.
Capacity Conversions
| Volume | Equals |
|---|---|
| 1 cubic metre | 1,000 litres |
| 1 litre | 1,000 cm³ |
| 1 US gallon | 3.785 litres |
| 1 imperial gallon | 4.546 litres |
| 1 cubic foot | 28.317 litres |
| 1 cubic yard | 764.6 litres |
A convenient physical anchor: one litre of water has a mass of one kilogram, and one cubic metre of water weighs one tonne.
Where Volume Calculations Matter
Concrete and aggregate ordering, tank and pool capacity, shipping and packaging, HVAC sizing by room volume, and dosing by container size. In construction the usual practice is to compute the volume then add 5–10% for waste, since running short mid-pour is far more expensive than a small surplus.
Frequently Asked Questions
How do I find the volume of an irregular object?
Water displacement. Submerge it in a measured container and the rise in level equals the object's volume — the method Archimedes is said to have discovered in a bath.
Why do gallons differ between countries?
The US gallon derives from the old English wine gallon of 231 cubic inches; the imperial gallon was redefined in 1824 as the volume of ten pounds of water. They differ by about 20%.
What is the difference between volume and capacity?
Volume is the space an object occupies; capacity is what a container can hold. The distinction matters when walls have thickness — external volume exceeds internal capacity.