Geometry formulas explained
Geometry comes down to a handful of formulas for perimeter, area, and volume, plus the Pythagorean theorem tying right triangles together. This guide explains what each formula measures and why it works, so the shapes make sense rather than needing pure memorization.
Perimeter, area, and volume
These three ideas measure different things. Perimeter is the distance around a flat shape, in plain units like metres. Area is the surface a flat shape covers, in square units. Volume is the space a solid fills, in cubic units. Keeping the dimensions straight, length for perimeter, length squared for area, length cubed for volume, is half the battle.
Area of common shapes
Most area formulas trace back to the rectangle, whose area is length times width. A triangle is half a rectangle of the same base and height, which is where its one-half comes from. A circle introduces pi, the ratio of circumference to diameter.
| Shape | Area |
|---|---|
| Rectangle | length × width |
| Triangle | ½ × base × height |
| Parallelogram | base × height |
| Trapezoid | ½(a + b) × height |
| Circle | π × radius² |
Volume of common solids
Volume formulas follow a similar logic. A box is its three dimensions multiplied together. A prism or cylinder is its base area times its height. The pointed solids, cones and pyramids, hold exactly one third of the prism or cylinder that encloses them.
| Solid | Volume |
|---|---|
| Rectangular box | length × width × height |
| Cylinder | π × radius² × height |
| Cone | (1/3) × π × radius² × height |
| Sphere | (4/3) × π × radius³ |
| Pyramid | (1/3) × base area × height |
Circles and pi
Every circle formula revolves around pi, about 3.14159, the constant ratio of a circle circumference to its diameter. The circumference is pi times the diameter, or two pi times the radius. The area is pi times the radius squared. Because area depends on the radius squared, doubling a circle radius quadruples its area.
The Pythagorean theorem
For any right triangle, the square of the longest side, the hypotenuse, equals the sum of the squares of the other two sides: a² + b² = c². This single relationship lets you find a missing side whenever a triangle has a right angle, and it underlies distance calculations, construction, and navigation. A triangle with sides 3, 4, and 5 is the classic example, since 9 plus 16 equals 25.
- Perimeter is in units, area in square units, volume in cubic units.
- Most area formulas come from the rectangle; a triangle is half of one.
- A prism or cylinder volume is base area times height; cones and pyramids are one third of that.
- Circle formulas all use pi, with area depending on the radius squared.
- The Pythagorean theorem, a² + b² = c², links the sides of any right triangle.
Geometry references
See the Area Formula Chart, Volume Formula Chart, and Circle Formula Chart.
FAQ
What is the difference between area and perimeter?
Perimeter is the distance around a shape, in plain units; area is the surface it covers, in square units.
Why does a triangle area have a one-half?
A triangle is exactly half of a rectangle with the same base and height, so its area is half of base times height.
What is the Pythagorean theorem used for?
Finding a missing side of a right triangle and measuring straight-line distances, using a² + b² = c².