Triangle Height Calculator — All Three Altitudes
Enter the three side lengths of any triangle to instantly find all three altitudes (heights) and the area. Uses Heron's formula to compute area from sides alone — no angle needed.
Perpendicular distance from vertex A to side a
- 1
Semi-perimeter
(5 + 6 + 7) ÷ 2 = 9 - 2
Heron's product
9 × 4 × 3 × 2 = 216The expression under the square root in Heron's formula. - 3
Area
√216 = 14.6969 - 4
Height h_a
2 × 14.6969 ÷ 5 = 5.8788
How does this calculator work?
Enter three side lengths; the calculator finds the area via Heron's formula (Area = √[s(s−a)(s−b)(s−c)]), then divides by each base: h_a = 2·Area/a, h_b = 2·Area/b, h_c = 2·Area/c. All three altitudes are returned instantly alongside the area and perimeter.
Formula
How this is calculated
Every triangle has three altitudes — the perpendicular distance from each vertex to the opposite side (or its extension). To find them from side lengths alone, the calculator first computes the area using Heron's formula: Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 is the semi-perimeter. Once the area is known, each altitude follows from the basic area formula: Area = ½ × base × height, rearranged as h = 2·Area / base. For example, h_a is the altitude dropped to side a, h_b to side b, and h_c to side c.
The three altitudes meet at a single interior point called the orthocentre for acute triangles (inside the triangle), and at a point outside for obtuse triangles. Heron's formula is valid for any triangle provided the three sides satisfy the triangle inequality — if any single side equals or exceeds the sum of the other two, no real triangle exists and the result is blank.
The method is exact up to floating-point arithmetic and does not require any angle input, making it useful when only side lengths are measured directly (for example, from a survey or a physical model).
Frequently asked questions
Only for acute triangles. In an obtuse triangle the altitude from the obtuse vertex falls outside the triangle (on the extension of the opposite side). The formula h = 2·Area / base still gives the correct perpendicular length regardless.
Yes — compute the area first (Area = ½ a b sin C) and then use h = 2·Area / base. For three sides, Heron's formula gives the area without needing any angle.
The three sides do not form a valid triangle — either a value is non-positive or the triangle inequality fails (one side ≥ sum of the other two). Adjust the sides so that the sum of any two exceeds the third.
Also known as
TG we-Calculate Editorial Team. (2026). Triangle Height Calculator — All Three Altitudes [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/triangle-height-calculator
TG we-Calculate Editorial Team. "Triangle Height Calculator — All Three Altitudes." TG we-Calculate. 2026. https://we-calculate.com/calculator/triangle-height-calculator.
TG we-Calculate Editorial Team, "Triangle Height Calculator — All Three Altitudes," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/triangle-height-calculator
@misc{wecalculate_triangle_height_calculator, title = {Triangle Height Calculator — All Three Altitudes}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/triangle-height-calculator}}, year = {2026}, note = {TG we-Calculate} }
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