Triangle Height Calculator

Enter a triangle's three side lengths to get its area (Heron's formula) and the height (altitude) relative to each side, h = 2A ÷ base.

Quick Facts

Semi-perimeter
s = (a + b + c) / 2
Half the triangle's total perimeter; feeds directly into Heron's formula.
Area (Heron's formula)
A = √(s(s−a)(s−b)(s−c))
Finds the area from three side lengths alone — no angle needed.
Height formula
h = 2A ÷ base
Each side has its own matching altitude; all three give the same area.

Your Results

Calculated
Area
-
A = √(s(s−a)(s−b)(s−c))
Height to Side a
-
h_a = 2A ÷ a
Height to Side b
-
h_b = 2A ÷ b
Height to Side c
-
h_c = 2A ÷ c

Ready

Enter the three side lengths, then press Calculate.

Formula and Method for Triangle Height

The height (or altitude) of a triangle is the perpendicular distance from one vertex to the line containing the opposite side (its base). Every triangle has three altitudes — one for each side — and each altitude pairs with its base to give the same area through the familiar formula A = ½ × base × height. When you only know the three side lengths, you cannot measure the perpendicular directly, so this calculator first finds the area with Heron's formula and then solves each height from that area.

How the calculation works

Given side lengths a, b, and c, the calculator computes the semi-perimeter s = (a + b + c) / 2, then the area with Heron's formula: A = √(s(s−a)(s−b)(s−c)). Once the area is known, the height relative to any side is found by rearranging the standard area formula: since A = ½ × base × height, the height is h = 2A ÷ base. Applying that to each side in turn gives h_a = 2A ÷ a, h_b = 2A ÷ b, and h_c = 2A ÷ c. Enter all three sides in the same unit — mixing units (e.g., feet with inches) produces an incorrect area and incorrect heights.

Common mistakes

  • Ignoring the triangle inequality: three lengths only form a triangle if each side is shorter than the sum of the other two (a + b > c, and so on). If this fails, no valid area or height exists.
  • Mixing units: keep all three side lengths in the same unit before entering them — convert inches to feet, or centimeters to meters, first.
  • Assuming one height fits all sides: the altitude is specific to whichever side you treat as the base. A right triangle's two legs can serve as base and height directly, but for any other base you need Heron's formula.

Real-world applications

  • Carpentry and roofing use triangle height to size rafters, gables, and roof trusses from measured edge lengths.
  • Land surveying and civil engineering compute triangular plot or parcel areas from boundary measurements, then derive heights for grading and drainage planning.
  • Structural and mechanical design use triangle altitudes to check clearances and load paths in triangulated frames and trusses.
  • Trigonometry and geometry coursework use this exact Heron's-formula-then-altitude method to solve "find the height" problems given only side lengths.

Frequently Asked Questions

How do you find the height of a triangle from its three sides?
First use Heron's formula to find the area: with semi-perimeter s = (a+b+c)/2, the area is A = √(s(s−a)(s−b)(s−c)). Then, for whichever side you treat as the base, the height to that base is h = 2A ÷ base.
What is Heron's formula?
Heron's formula finds a triangle's area from its three side lengths alone, with no need to know any angle or height directly: A = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter.
Does a triangle have more than one height?
Yes. Every triangle has three altitudes, one perpendicular to each side (base). They are usually different lengths, but each one produces the same area when combined with its matching base: A = ½ × base × height.
Why does my calculator say the side lengths cannot form a triangle?
The triangle inequality theorem requires each side to be shorter than the sum of the other two sides. If a + b ≤ c (or any similar combination), the three lengths cannot close into a triangle, so no area or height exists.

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