Geometry
Area from the three sides (Heron's formula)
Use when no height and no angle are available. Halve the perimeter first, then subtract each side from that half-sum before multiplying.
The Formula
A = sqrt(s(s - a)(s - b)(s - c)), where s = (a + b + c) / 2
Use when no height and no angle are available. Halve the perimeter first, then subtract each side from that half-sum before multiplying.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| s | Semi-perimeter | length | Half the perimeter. It is s, not the full perimeter, that enters every one of the four factors. |
| a | Length of one side | length | Together with b and c, it must satisfy the triangle inequality or no triangle exists. |
| A | Enclosed area | square units | For sides of 7, 8 and 9 this is sqrt(720) = 26.83281573. |
What Each Symbol Means
slength
Semi-perimeter
Half the perimeter. It is s, not the full perimeter, that enters every one of the four factors.
alength
Length of one side
Together with b and c, it must satisfy the triangle inequality or no triangle exists.
Asquare units
Enclosed area
For sides of 7, 8 and 9 this is sqrt(720) = 26.83281573.
See It Worked With Real Numbers
A formula on its own does not tell you which number goes where. The how to calculate the area of a triangle page applies this equation to a concrete case, shows every intermediate step, and names the mistakes that produce a wrong answer even when the algebra is right.
Where This Formula Appears
This equation is the working core of How To Calculate The Area Of A Triangle, filed under Math. Every number the guide produces comes from substituting into this one line.