By Damon Verial Updated Aug 30, 2022
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Calculating the height of a triangle can seem tricky because each triangle is unique. However, by leveraging the data you have—such as area, side lengths, or angles—you can choose the most appropriate method. Below we outline four reliable techniques, complete with step‑by‑step explanations and practical examples.
Sometimes you know the area and base of a triangle but not its height. You can manipulate the area formula, A = ½ b h, to isolate the height: h = 2A / b. For example, if the area is 36 and the base is 9, then h = 2 × 36 ÷ 9 = 8.
If you know the base and the length of one other side (the hypotenuse) of a right triangle, you can find the height by drawing a perpendicular from the vertex to the base. This creates a right triangle inside the original. Apply a² + b² = c², where b is the base and c is the hypotenuse. For a base of 3 and a hypotenuse of 5, we get a² + 9 = 25, so a² = 16 and a = 4.
When you know an angle adjacent to the height and the hypotenuse, the tangent function is useful. Set tan(θ) = h / (b / 2). For an angle of 30° and a base of 6, tan(30°) = h / 3. Solving gives h = 3 × tan(30°) = 3 × (√3 / 3) = √3.
Heron's formula first finds the half‑perimeter s = (a + b + c) / 2, then the area A = √[s (s – a)(s – b)(s – c)]. Once you have A, compute the height with h = 2A / b. For sides 6, 8, 10, we get s = 12 and A = √[12 × 6 × 4 × 2] = √576 = 24. With a base of 10, h = 2 × 24 ÷ 10 = 4.8.
