Question

What is the formula for finding the area of a quadrilateral?

Answer 1

If a convex quadrilateral is defined by all its sides and all its interior angles, it can be broken into two triangles and the area can be expressed as a sum of the areas of these triangles.

Explanation:

Assume in a convex quadrilateral `ABCD` we know all its sides and all its interior angles:
`AB = a`
`BC = b`
`CD = c`
`DA = d`
`/_BAD = alpha`
`/_ABC = beta`
`/_BCD = gamma`
`/_CDA = delta`

Draw a diagonal `AC`. It divides our quadrilateral into two triangles, and for each one of them we know two sides and an angle between them.

For triangle `Delta ABC` we know
`AB = a`
`BC = b`
`/_ABC = beta`

Taking `a` as a base of `Delta ABC`, the altitude would be `b*sin(beta)`.
The area of this triangle is
`S_(ABC) = 1/2*a*b*sin(beta)`

For triangle `Delta ADC` we know
`AD = d`
`CD = c`
`/_ADC = delta`

Taking `c` as a base of `Delta ADC`, the altitude would be `d*sin(delta)`.
The area of this triangle is
`S_(ADC) = 1/2*c*d*sin(delta)`

The total area of a quadrilateral is, therefore,
`S = 1/2[a*b*sin(beta)+c*d*sin(delta)`]