Formulas & Equations

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This is an incomplete list of formulas used in geometry.

1 Length
- 1.1 Perimeter and Circumference
  - 1.1.1 Polygon
  - 1.1.2 Circle
- 1.2 Triangles
2 Area
3 Volume

Length

Perimeter and Circumference

Polygon

Sum the lengths of the sides.

Circle

- $d\,$ is the diameter

Triangles

Pythagorean theorem: In a right triangle where $a\,$ and $b\,$ are the lengths of the legs and $c\,$ is the length of the hypotenuse, $a^2+b^2=c^2\,$

Area

Triangles

- $b\,$ = base, $h\,$ = height (perpendicular to base)
Heron's Formula:
- $a, b, c\,$ are sides, and $s = \frac{a+b+c}{2} \,$

Equilateral Triangles

- $a\,$ is a side

Quadrilaterals

Squares

- $s\,$ is the length of the square's side

Rectangles

- $a\,$ and $b\,$ are the sides of the rectangle

Parallelograms

- $b\,$ is the base, $h\,$ is the height

Trapezoids

- $b_1,b_2\,$ are the two bases, $h\,$ is the height

Circles

- $r\,$ is the radius

Surface Areas

Cube: 6×( $s^2$ )
- $s\,$ is the length of a side.
Rectangular Prism: 2×(( $l,$ × $w\,$ ) + ( $l\,$ × $h\,$ ) + ( $w\,$ × $h\,$ ))
- $l\,$ , $w\,$ , and $h\,$ are the length, width, and height of the prism
Sphere: 4×π×( $r\,$ ²)
- $r\,$ is the radius of the sphere
Cylinder: 2×π× $r\,$ ×( $h\,$ + $r\,$ )
- $r\,$ is the radius of the circular base, and $h\,$ is the height
Pyramid:
- $A$ = Surface area, $A_b$ = Area of the Base, $p$ = Perimeter of the base, $s$ = slant height.

The surface area of a regular pyramid can also be determined based only on the number of sides( $n$ ), the radius( $r$ ) or side length( $l$ ), and the height( $h$ )

If $r$ is known, $l$ is defined as $l = \sqrt{(rcos(\frac{360}{n})-r)^2 + (rsin(\frac{360}{n}))^2} = \sqrt{2}r\sqrt{1-cos(\frac{360}{n})}$

or if $l$ is known, $r$ is defined as $r = \frac{l}{\sqrt{2}\sqrt{1-cos(\frac{360}{n})}}$

The slant height $h_1$ is given by $\sqrt{r^2+h^2+\frac{l^2}{4}}$

The total surface area of the pyramid is given by $n\frac{l}{2}[h_1 + h_0]$

Cone: π×r×(r + √(r² + h²))
- $r\,$ is the radius of the circular base, and $h\,$ is the height.

Volume

Cube
- s = length of a side
Rectangular Prism
- l = length, w = width, h = height
Cylinder(Circular Prism)
- r = radius of circular face, h = distance between faces
Any prism that has a constant cross sectional area along the height:
- $A \cdot h$
- A = area of the base, h = height
Sphere:
- r = radius of sphere
Ellipsoid:
- a, b, c = semi-axes of ellipsoid
Pyramid:
- A = area of base, h = height from base to apex
Cone (circular-based pyramid):
- r = radius of circle at base, h = distance from base to tip