What is a triangle altitude?
An altitude of a triangle is a perpendicular segment dropped from a vertex to the line containing the opposite side. Every triangle has three altitudes, one to each side. This calculator finds all three heights (\(h_a\), \(h_b\), \(h_c\)) directly from the three side lengths, along with the triangle's area.
How to use it
Enter the three side lengths a, b and c in any consistent unit. The calculator first computes the area with Heron's formula, then divides twice the area by each side to obtain the altitude that lands on that side. Make sure the three sides form a valid triangle (each side shorter than the sum of the other two).
The formula explained
The area of a triangle equals one-half base times height, so \(\text{Area} = \tfrac{1}{2}\cdot a\cdot h_a\). Solving for the height gives \(h_a = 2\cdot\text{Area} / a\). Because the same area applies to every side, \(h_b = 2\cdot\text{Area} / b\) and \(h_c = 2\cdot\text{Area} / c\). The area itself comes from Heron's formula, $$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)},$$ where \(s = (a+b+c)/2\) is the semi-perimeter.
Worked example
For a 3-4-5 right triangle: \(s = (3+4+5)/2 = 6\), $$\text{Area} = \sqrt{6\cdot 3\cdot 2\cdot 1} = \sqrt{36} = 6.$$ Then \(h_a = 2\cdot 6/3 = 4\), \(h_b = 2\cdot 6/4 = 3\), and \(h_c = 2\cdot 6/5 = 2.4\). The altitudes to the legs equal the other leg, as expected for a right triangle.
FAQ
Do longer sides have shorter altitudes? Yes — since the area is fixed, the altitude is inversely proportional to its side, so the longest side has the shortest altitude.
What if my sides don't form a triangle? If the value inside the square root is zero or negative the area is reported as 0, indicating a degenerate or impossible triangle.
What units does it use? Any unit you like — the altitudes come out in the same linear unit as the sides, and the area in that unit squared.
