A four-sided plot has sides of 15 m, 10.2 m, 4.5 m and 15.3 m. Can its area be found from the side lengths alone?

Step-by-step solution

1. 1

   The four sides are not enough. Unlike a triangle, a quadrilateral is not rigid: you can hinge it at the vertices and flex it. The same four side lengths bound infinitely many different shapes with different areas, so no formula can return a single answer from sides alone.

2. 2

   Identify what one extra measurement would fix it. Any one of these pins the shape down: a diagonal, one interior angle, the height, or the knowledge that the shape is a specific type (trapezoid, parallelogram, cyclic).

3. 3

   With a diagonal, split into two triangles. Draw the diagonal $d$ between the $15$ and $4.5$ vertices. That gives a triangle with sides $15$, $10.2$, $d$ and another with sides $4.5$, $15.3$, $d$.

4. 4

   Apply Heron's formula to each. For a triangle with sides $a, b, c$ and semi-perimeter $s = \tfrac{a+b+c}{2}$,

   $A = \sqrt{s(s-a)(s-b)(s-c)}$

   Then the quadrilateral's area is $A = A_1 + A_2$.

5. 5

   Special case worth knowing. If the quadrilateral is cyclic (all four vertices on one circle) then the sides do determine the area, via Brahmagupta's formula:

   $A = \sqrt{(s-a)(s-b)(s-c)(s-d)}$

   This is the maximum area achievable with those four sides.