By Chris Deziel
Apr 14, 2023 2:40 am EST
wutwhanfoto/iStock/GettyImages
A scalene triangle has three unequal sides and three distinct angles. Unlike equilateral, isosceles, or right triangles, its dimensions are not symmetrical, which means its area can’t be found with a single universal shortcut. However, with a few measurements you can determine its area accurately using classical geometry.
Choose any side as the base (denoted b) and draw the altitude from the opposite vertex. The altitude is the perpendicular distance to the base (denoted h). The triangle’s area is then simply half the product of base and height:
\[\text{Area} = \tfrac{1}{2}\,b\,h\]
This formula works for every triangle, but finding the exact height can be tricky, especially for obtuse triangles where the altitude falls outside the triangle’s interior.
When you have the lengths of all three sides (a, b, and c), Heron’s formula lets you compute the area without needing a height. First calculate the semi‑perimeter:
\[s = \tfrac{1}{2}(a + b + c)\]
Then the area follows:
\[\text{Area} = \sqrt{s\,(s-a)\,(s-b)\,(s-c)}\]
Heron’s formula is reliable for any triangle, including scalene, equilateral, and isosceles shapes.
If you know two sides and the angle they form, you can first compute the third side using the Law of Cosines:
\[c^2 = a^2 + b^2 - 2ab\cos C\]
After determining the missing side, plug all three side lengths into Heron’s formula to obtain the area. This method is useful when a direct height measurement is unavailable but an angle and two adjacent sides are known.
