What is Heron's formula for the area of a triangle?

Heron's formula calculates the area of a triangle from its three side lengths (a, b, c) without needing an angle or the height. First compute the semi-perimeter: s = (a + b + c) / 2. Then the area is: Area = sqrt(s x (s - a) x (s - b) x (s - c)). The formula works for any valid triangle, including scalene, isosceles, and equilateral triangles.

When does a set of three sides NOT form a valid triangle?

Three sides form a valid triangle only when each side is shorter than the sum of the other two. This is called the triangle inequality. For example, sides of 1, 2, and 10 do not form a triangle because 1 + 2 = 3, which is less than 10. If this condition is violated, the value under the square root in Heron's formula becomes negative, indicating the sides cannot close into a triangle. This calculator checks the triangle inequality and shows an error if the sides are invalid.

How is Heron's formula different from the base-times-height formula?

The standard formula Area = 0.5 x base x height requires you to know the perpendicular height, which can be difficult to measure directly on an irregular triangle. Heron's formula only needs the three side lengths, making it ideal when you can measure all sides but not the height. Both formulas give the same result. For a right-angled triangle, the two legs serve as base and height, so the standard formula is simpler. For scalene or obtuse triangles where the height falls outside the triangle, Heron's formula is often more convenient.

Area of a Triangle (Heron's Formula) Calculator

Find the area of any triangle using only its three side lengths. Enter sides a, b, and c and the calculator applies Heron's formula instantly. No height or angle needed.

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Standard Formula Heron's formula is an exact result from Euclidean geometry. No expiry.

Triangle Side Lengths

Side a (any unit)

units

Side b (any unit)

units

Side c (any unit)

units

Unit label (optional, for display)

Formula Reference

Heron's formula requires only the three side lengths:

s = (a + b + c) / 2

Area = √[s × (s − a) × (s − b) × (s − c)]

where s is the semi-perimeter. The result is in square units of your side lengths.

The three sides must satisfy the triangle inequality: each side must be less than the sum of the other two.

Results

Area
-
square units

Semi-perimeter (s)

units

Perimeter

units

Triangle Type

by sides

Step-by-Step Working

Side a-

Side b-

Side c-

s = (a + b + c) / 2-

s − a-

s − b-

s − c-

s(s-a)(s-b)(s-c)-

Area = √(product)-

Triangle Properties

Perimeter (a + b + c)-

Semi-perimeter-

Longest side-

Triangle inequality-

Triangle type (sides)-

Angle type-

Area-

Result: Enter three positive side lengths above.

How Heron's Formula Works

Heron's formula (named after Hero of Alexandria, c. 60 AD) gives the area of any triangle directly from its three side lengths. The first step is to find the semi-perimeter, which is half the triangle's total perimeter:

s = (a + b + c) / 2

The area is then:

Area = √[s × (s − a) × (s − b) × (s − c)]

If all three sides are equal (an equilateral triangle), the formula simplifies to Area = (√3 / 4) × a², but Heron's formula handles every triangle type equally well.

Worked Example

Using the default values: a = 5, b = 6, c = 7.

Step	Calculation	Result
Semi-perimeter s	(5 + 6 + 7) / 2	9
s − a	9 − 5	4
s − b	9 − 6	3
s − c	9 − 7	2
Product	9 × 4 × 3 × 2	216
Area	√216	14.6969 square units

The perimeter is 5 + 6 + 7 = 18 units. Because all three sides are different lengths, the triangle is scalene.

Triangle Types by Side Length

Type	Condition	Example
Equilateral	a = b = c	3, 3, 3
Isosceles	Exactly two sides equal	5, 5, 8
Scalene	All three sides different	5, 6, 7

Triangle Inequality

Three lengths can only form a triangle if each side is strictly less than the sum of the other two. For sides a, b, c:

a + b > c
a + c > b
b + c > a

If any condition fails, the three lengths cannot close into a triangle. In Heron's formula this appears as a negative value under the square root, which has no real solution.

Angle Type from Side Lengths

You can determine whether a triangle is acute, right-angled, or obtuse by comparing the square of the longest side to the sum of squares of the other two. Let c be the longest side:

If c² < a² + b²: acute triangle (all angles less than 90 degrees)
If c² = a² + b²: right-angled triangle (Pythagorean theorem)
If c² > a² + b²: obtuse triangle (one angle greater than 90 degrees)

Related Calculators

Maths and Stats Calculators: full index of geometry and maths tools.
Area of a Triangle Calculator: area from base and height, or with an angle.
Triangle Calculator: solve any triangle from sides and angles.
Pythagorean Theorem Calculator: find a missing side in a right triangle.
Area Calculator: area formulas for circles, rectangles, and more.

Sources and method: Heron of Alexandria, Metrica (c. 60 AD). The formula is a standard result of Euclidean plane geometry and appears in all major mathematics references including the NZQA Level 2 Mathematics curriculum (AS91259).

This calculator applies Heron's formula exactly. Results are accurate to the precision of your inputs. The tool checks the triangle inequality before computing and will show an error if the three sides cannot form a valid triangle.

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