Geometry Formulas: Area, Volume, Perimeter

The formula for every standard shape, where it comes from, and worked examples

Geometry Formulas: Area, Volume, Perimeter

Formula:

Every area, volume and perimeter formula in one place

These are the formulas for the standard shapes: area, perimeter, circumference, diameter, volume and surface area. Each formula has its own page, and on it there is not only the formula but where it comes from, what every letter means, two worked examples and the mistake that this particular shape invites.

A formula can be found anywhere. What is harder to find is why it is that formula — why the cone carries a third, where πr² comes from, why the semicircle adds the diameter to the arc. That is what the pages are written for.

Area

Shape Formula
CircleA = π · r²
RectangleA = a · b
SquareA = a²
TriangleA = ½ · a · h
TrapezoidA = ½ · (a + b) · h
ParallelogramA = a · h
RhombusA = ½ · d · e
HexagonA = (3√3 / 2) · a²
Regular polygonA = (n · a²) / (4 · tan(π / n))
Circular sectorA = (deg / 360) · π · r²

Perimeter

Shape Formula
RectangleP = 2 · (a + b)
SquareP = 4 · a
TriangleP = a + b + c
Regular polygonP = n · a
SemicircleP = π · r + 2 · r

Circumference

Shape Formula
CircleC = 2 · π · r

Diameter

Shape Formula
Circled = 2 · r

Volume

Shape Formula
ConeV = (1/3) · π · r² · h
CylinderV = π · r² · h
SphereV = (4/3) · π · r³
CubeV = a³
Rectangular prismV = a · b · h
Triangular prismV = ½ · a · h · l
PyramidV = (1/3) · a · b · h

Surface area

Shape Formula
CylinderS = 2 · π · r² + 2 · π · r · h
SphereS = 4 · π · r²
CubeS = 6 · a²
Rectangular prismS = 2 · (a·b + a·h + b·h)
ConeS = π · r² + π · r · l
PyramidS = a² + 2 · a · l
Triangular prismS = a · h + 3 · a · l

What is not here, and why

A flat figure has no volume and no surface area distinct from its area, so there are no «volume of a circle» or «surface area of a rectangle» pages here, even though people do type those. A separate address for them would print an error of subject as a fact; instead they are dealt with inside the neighbouring pages.

There is no separate page for the perimeter of a circle either — not because nobody asks for it, but because it is the same length as the circumference. One value, one address.

See Also

Geometry Formulas: Area, Volume, Perimeter

These are the formulas for the area, perimeter, circumference, volume and surface area of the standard shapes — 31 of them. Each page shows not only the formula but where it comes from, what every letter stands for, two worked examples and the mistake that this particular shape invites. For example, it can help you find out what is the formula for the area of a circle? (The answer is: A = π · r²). Pick the pair — "Area of a circle", for instance — and hit the "Find out" button.

FAQ

What is the formula for the area of a circle?

A = πr², where r is the radius. The most common mistake is putting the diameter in place of the radius: because the radius is squared, that makes the answer four times too big. A circle 10 units across has area π·5² ≈ 78.54, not π·10².

What is the difference between area and perimeter?

The perimeter is the length of the outline and comes out in units; the area is how much surface is enclosed and comes out in square units. For a rectangle both formulas take the same two numbers — 2(a + b) and a·b — which is exactly why they get swapped.

Why is the volume of a cone one third of a cylinder?

Because it measures out that way: fill a cone with water and it takes exactly three of them to fill a cylinder of the same base and the same height. The one third is exact, not an approximation, and the same factor appears in the pyramid formula for the same reason.

Does a circle have a perimeter?

Yes, and it has a name of its own: the circumference, C = 2πr. «Perimeter of a circle» and «circumference of a circle» are two names for the same length, so there is one formula and not two.