Pythagorean theorem
Calculate the hypotenuse or a missing leg of a right triangle from the lengths of the other two sides.
c² = a² + b² Calculated quantities
a - first leg · b - second leg · c - hypotenuse
Math formulas
Geometry formulas
Use geometric relationships to calculate lengths, areas, angles, and other properties of plane figures and solids.
Formulas and calculators
Open formula
Circle area and circumference
Calculate radius, diameter, circumference, or area of a circle from one known measurement.
A = πr²; C = 2πr Calculated quantities
r - radius · d - diameter · C - circumference · A - area
Triangle area
Calculate triangle area from a base and perpendicular height or from three side lengths using Heron’s formula.
A = b × h / 2 Calculated quantities
A - area · b - base · h - perpendicular height · a - first side · b - second side · c - third side
Rectangle area and perimeter
Calculate rectangle area, perimeter, or a missing side from the available dimensions.
A = a × b; P = 2(a + b) Calculated quantities
A - area · P - perimeter · a - first side · b - second side
How to choose the right formula
Pythagorean theorem
c² = a² + b² - When to use it
- In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs.
- Calculated quantities
- a - first leg (m) · b - second leg (m) · c - hypotenuse (m)
- Worked example: Hypotenuse of a 3-4-5 triangle
- The legs of a right triangle are 3 m and 4 m long. Calculate the hypotenuse.
Circle area and circumference
A = πr²; C = 2πr - When to use it
- The circle area is π times the radius squared, while its circumference is 2π times the radius.
- Calculated quantities
- r - radius (m) · d - diameter (m) · C - circumference (m) · A - area (m²)
- Worked example: Area from a five-centimetre radius
- A circle has a radius of 5 cm. Calculate its area.
Triangle area
A = b × h / 2 - When to use it
- Triangle area is one half of a chosen base multiplied by the perpendicular height to that base.
- Calculated quantities
- A - area (m²) · b - base (m) · h - perpendicular height (m) · a - first side (m) · b - second side (m) · c - third side (m)
- Worked example: Area from base and height
- A triangle has a base of 10 cm and a perpendicular height of 6 cm. Find its area.
Rectangle area and perimeter
A = a × b; P = 2(a + b) - When to use it
- Area is the product of adjacent side lengths, while perimeter is twice their sum.
- Calculated quantities
- A - area (m²) · P - perimeter (m) · a - first side (m) · b - second side (m)
- Worked example: Area of an 8 by 5 rectangle
- A rectangle has sides of 8 cm and 5 cm. Calculate its area.
What to check before calculating
Assumptions and conditions
- The triangle is a plane right triangle in Euclidean geometry.
- Side c is the hypotenuse opposite the right angle; a and b are the legs.
- All side lengths are positive and must use compatible units.
- When calculating a leg, the hypotenuse must be longer than the known leg.
- The figure is a circle in Euclidean plane geometry.
- Radius, diameter, and circumference must be positive lengths.
- The area uses squared units; a length unit cannot be used as an area unit.
- Results containing π are exact before the displayed decimal approximation is rounded.
Common mistakes
- Using the theorem for a triangle that is not right-angled.
- Treating a leg as the hypotenuse. The hypotenuse must be opposite the right angle and must be the longest side.
- Adding side lengths instead of their squares.
- Forgetting to take the square root after adding or subtracting the squares.
- Confusing radius with diameter.
- Using A = 2πr or C = πr².
- Forgetting to square the area unit.
- Rounding π too early in a multi-step calculation.