Geometric Shape Calculator

Geometric formulas and the scaling rules behind them

Every area and volume formula in this tool follows two rules that are worth understanding once rather than memorising thirty times: area scales with the square of length, and volume scales with the cube. That single fact explains most of what looks surprising about shapes.

The square-cube law, and why it matters outside maths

Double every dimension of an object and its surface area quadruples while its volume increases eightfold. The ratio of surface to volume therefore halves.

This governs a surprising range of real behaviour. Small animals lose heat quickly because they have a large surface relative to their mass. Large animals need disproportionately thick legs, because weight grows with volume while bone strength grows with cross-sectional area. Crushed ice melts faster than a single block of the same mass for the same reason.

In engineering terms, it means you cannot scale a design up by simply multiplying every dimension. A model bridge at 1:100 that holds its own weight tells you nothing about whether the full-size version will.

Perimeter, area, and volume are independent

Knowing one does not give you the others. Two shapes with identical perimeters can have very different areas, and the same holds for surface area against volume.

For a fixed perimeter, area is maximised by the circle. For a fixed surface area, volume is maximised by the sphere. This is the isoperimetric principle, and it is why soap bubbles are spherical and why storage tanks tend toward cylinders with roughly equal height and diameter.

The practical version: order fencing by perimeter, turf by area, and soil by volume. No one of those measurements can be derived from another.

Formulas grouped by what you are measuring

• Two-dimensional area: circle pi r^2, rectangle l x w, triangle b x h / 2, trapezium (a + b) x h / 2, regular polygon perimeter x apothem / 2.
• Perimeter: circle 2 pi r, rectangle 2(l + w), triangle a + b + c.
• Volume: cube a^3, cuboid l x w x h, cylinder pi r^2 h, cone pi r^2 h / 3, sphere 4/3 pi r^3, pyramid base area x h / 3.
• Surface area: cube 6a^2, cylinder 2 pi r(r + h), sphere 4 pi r^2, cone pi r(r + slant height).
• Note the pattern: cones and pyramids are exactly one third of the prism that contains them, regardless of base shape.

Worked example

A cylindrical water tank is 1.2 m in diameter and 1.8 m tall. How much does it hold, and how much steel does it need?

Volume: r = 0.6, so V = pi x 0.6^2 x 1.8 = 2.036 m^3, which is 2,036 litres.

Surface area including both ends: 2 pi r (r + h) = 2 x pi x 0.6 x (0.6 + 1.8) = 9.05 m^2.

Now scale the tank up by 25% in every dimension. Volume rises by 1.25^3 = 1.95, nearly double, to 3,975 litres. Surface area rises by only 1.25^2 = 1.56, to 14.1 m^2. The larger tank holds 95% more water for 56% more steel - the square-cube law working in your favour.

What happens when you scale a shape by factor k

Frequently Asked Questions

Related calculators

• Circle Area Calculator
• Rectangle Area Calculator
• Triangle Area Calculator
• Volume Converter

References

• NIST Digital Library of Mathematical Functions
• NIST Special Publication 811 - Guide for the Use of the SI

Last reviewed: 2026-08-07. This page is informational. For legal, medical, tax, or financial decisions, confirm the result with a qualified professional.