# Quadrilateral

Calculates Convex quadrilateral area with Bretschneider's formula.

All below formulas apply to a convex quadrilateral.

## Quadrilateral area formulas

We denote (see diagram above) :

- a, b, c, d : lengths of the quadrilateral sides

- `A, B, C, D` : quadrilateral interior angles

- p, q : lengths of diagonals

- `\alpha` : angle formed by diagonals

**Area formula knowing sides lengths and the sum of 2 opposite angles**

We use Bretschneider's formula,

`Ar = sqrt((s-a)*(s-b)*(s-c)*(s-d)-a*b*c*d*cos^2((A+C)/2))`

s is the semi-perimeter, `s=(a+b+c+d)/2`

In this formula, we may replace `A+C` by `B+D` since `A+B+C+D = 360°` in a convex quadrilateral and then

`cos^2((A+C)/2) = cos^2((B+D)/2)`

**Area formula knowing sides lengths and diagonals lengths**

`Ar = 1/4*sqrt(4*p^2*q^2-(b^2+d^2-a^2-c^2)^2)`

**Area formula knowing diagonals lengths and their angle**

`Ar = 1/2*p*q*sin(\alpha)`

## See also

Plane Geometry calculators

Geometry calculators

Mathematics calculators