The formula
h = 2 × area ÷ base, with the area from Heron's formula
Where it comes from
A triangle has three heights, one to each side, and they are generally all different. Since the area can be computed from any base-height pair, knowing the area from the sides gives you every height at once.
How to work it out by hand
- Work out the semi-perimeter: the three sides added and halved
- Apply Heron's formula for the area
- Each height is twice the area divided by its own base
- Check: every base-height pair should give the same area
What is worth knowing
Heron's formula is remarkable for needing no angles at all — three lengths are enough for the area, which is not obvious and was not easy to prove. The three heights meet at a single point called the orthocentre, which unlike the centroid can fall outside the triangle entirely when one angle is obtuse. The heights are also inversely proportional to their bases, which follows immediately from the area being fixed: the longest side always has the shortest height.