Find trapezoid height if all sides are given

Question

What is the formula to find height of trapezoid if all sides are given?

For example if lengths of sides are:
a: 11, b: 3, c: 7, d: 6

How would we calculate length of height (h)?

What is the formula for this? trapezoid

Answer 1

Let $a$ and $c$ are bases, as in your picture. Then height of trapezoid is the same as height of triangle with sides $|a-c|$, $b$ and $d$. You can find the height of triangle from sides using two formulae of triangle area: $$S=\sqrt{p(p-|a-c|)(p-b)(p-d)}=\frac{h|a-c|}{2},$$ where $p=\frac{|a-c| +b+d}{2}$.