Math

Heron's Formula Calculator

Calculate Triangle Area from Three Side Lengths

Heron's formula computes the area of any triangle directly from its three side lengths, without needing any angle. Enter sides a, b, and c to compute the semiperimeter s and area A = √[s(s−a)(s−b)(s−c)].

Triangle Solver Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Right Triangle

Classic 3-4-5 right triangle using SSS mode.

Key values: a = 3 · b = 4 · c = 5

Roof Pitch

Calculate a roof triangle with two sides and a 35-degree pitch angle.

Key values: a = 6 · b = 8 · C = 35°

Equilateral Triangle

A perfect equilateral triangle with all sides equal to 10.

Key values: a = 10 · b = 10 · c = 10

Documentation

This calculator is also known as Heron's Formula Calculator.

Read the complete guide

Heron's Formula Derivation

Heron of Alexandria (c. 60 AD) showed that the area of a triangle with sides a, b, c and semiperimeter s = (a+b+c)/2 is: Area = √[s(s−a)(s−b)(s−c)]. The formula can be derived from the standard base-height formula by expressing the height in terms of the cosine rule.

Examples

Classic 3-4-5 Right Triangle

Find the area of a triangle with sides 3, 4, 5.

s = (3+4+5)/2 = 6. Area = √[6(6-3)(6-4)(6-5)] = √[6×3×2×1] = √36 = 6.

Key takeaway: For the 3-4-5 right triangle, ½bh = ½×3×4 = 6. Heron's formula confirms the same result.

Equilateral Triangle (side 10)

Find the area of an equilateral triangle with all sides = 10.

s = 15. Area = √[15×5×5×5] = √1875 ≈ 43.30.

Key takeaway: For an equilateral triangle with side a, Area = (√3/4)a². For a=10: (√3/4)×100 ≈ 43.30.

Scalene Triangle (land plot)

A triangular land plot has sides 150 m, 200 m, and 175 m. Find its area.

s = 262.5. Area = √[262.5×112.5×62.5×87.5] ≈ 12,830 m².

Key takeaway: Heron's formula is the standard method for calculating land areas when GPS coordinates give the fence post distances.

Using Heron's Formula Accurately

Avoid common mistakes:

Check the triangle inequality (a+b>c, a+c>b, b+c>a) before computing
The semiperimeter s is half the perimeter, not the perimeter itself
If s(s−a)(s−b)(s−c) is negative or zero, the sides don't form a valid triangle
Use sufficient decimal precision when sides are large to avoid rounding errors

Frequently Asked Questions about Heron's Formula Calculator

What is Heron's formula?

Heron's formula calculates triangle area from three side lengths: A = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 is the semiperimeter. It works for any triangle without needing any angle.

When should I use Heron's formula vs. ½bh?

Use Heron's formula when you know all three sides but not the height. Use ½bh when you know the base and perpendicular height directly. Both give the same answer.

Does Heron's formula work for right triangles?

Yes. For a right triangle with legs a, b and hypotenuse c, Heron's formula gives the same area as ½ab. For example, the 3-4-5 triangle: s=6, area = √(6×3×2×1) = 6 = ½×3×4.