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Heron's Formula Calculator

Find a triangle's area from three sides.

Result

26.83 units²
24.00 units

How it is calculated

From the three side lengths a, b and c (same unit) the tool first finds the semi-perimeter s = (a + b + c) / 2. The area follows from A = √( s(s − a)(s − b)(s − c) ). It also shows the perimeter a + b + c. Area is labelled units² and perimeter units; both have two decimals.

All sides must be positive. They must also satisfy the triangle inequality, meaning each side is shorter than the sum of the other two. Sides such as 3, 4 and 7 only lie flat on a line (3 + 4 equals 7), so the tool gives no result for them, and the same goes for 3, 4 and 8. The formula works for any triangle shape, right-angled or not, and the side order does not matter.

Worked example

Take sides 7, 8 and 9. Perimeter: 7 + 8 + 9 = 24, so s = 12. The differences are 12 − 7 = 5, 12 − 8 = 4 and 12 − 9 = 3. The product is 12 × 5 × 4 × 3 = 720, and √720 = 26.83. The area is 26.83 units² and the perimeter is 24.00 units.

Try this example →

Common mistakes

  • Entering a side length that equals or exceeds the sum of the other two; no triangle exists then and the tool returns nothing.
  • Using the perimeter instead of the semi-perimeter in the product, which gives an area that is far too large.
  • Entering a height or an angle in place of a side; Heron's formula needs three side lengths only.
  • Measuring the sides in different units, such as feet and inches; convert before entering.
  • Rounding the sides early on a thin triangle: when one side is almost the sum of the other two, small measurement errors change the area a lot.

About this calculator

This calculator finds a triangle's area using Heron's formula, Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter (a+b+c)/2 and a, b, c are the three side lengths. Attributed to Heron of Alexandria (1st century AD), its key advantage is that it needs only the three side lengths — no height or angle measurement required — making it the natural choice whenever a triangle's height isn't directly measurable.

Surveyors and land assessors use it to compute the area of irregularly shaped triangular plots measured only by boundary lengths; architects and civil engineers use it for triangular structural sections and site layouts; and geometry students use it as a standard alternative to the base-times-height formula. Enter the three side lengths to get the triangle's area.

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