Units are yours to choose
These formulas are unit-agnostic. Enter centimetres and areas come out in square centimetres; enter feet and you get square feet. The only rule is to use the same unit for every input in a single calculation.
Mixing units is the most common source of wrong answers — a cylinder with radius in centimetres and height in metres produces a volume that means nothing.
Area, surface area and volume
Area measures a flat region in square units. Surface area measures the total outside of a solid, also in square units — it is what you would need to paint or wrap. Volume measures the space inside, in cubic units.
Scaling reveals how they differ. Double every dimension of a solid and its surface area quadruples while its volume increases eightfold. This is why large animals have proportionally smaller surface areas and why crushed ice melts faster than a single block of the same mass.
Pi and the circular formulas
Pi is the ratio of any circle's circumference to its diameter, roughly 3.14159. Every circular and spherical formula here derives from it: circumference is 2πr, area is πr², sphere volume is 4/3 πr³.
The exponent tells you the dimension. Circumference is linear in r, area is quadratic, volume is cubic — which is why a modest increase in radius produces a dramatic increase in volume.
Frequently asked questions
Why does the three-sided triangle sometimes refuse to compute? +
Because of the triangle inequality: any side must be shorter than the sum of the other two. Sides of 1, 2 and 10 cannot close into a triangle.
How do I find the area of an irregular shape? +
Divide it into rectangles and triangles, compute each, and add them. For land in particular this is far more reliable than trying to fit a single formula.
What is the difference between a cone and a pyramid? +
A cone has a circular base, a pyramid a polygonal one. Both have volume equal to one third of base area times height.