Enter the three side lengths of any triangle to find its area with Heron's formula. You get the semi-perimeter, each step with your numbers, an exact square-root form when there is one, plus the angles, heights and a drawing to scale. Side lengths that cannot form a triangle are flagged.
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Enter your values to see the result.
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How to use the Heron's Formula Calculator
Enter the three side lengths a, b and c in the same unit.
Pick a unit (optional) so the area is shown in square units.
Press Calculate area; results also update as you type.
Read the area, semi-perimeter and the step-by-step working.
Check the drawing and table for the angles and heights.
Heron's formula explained
For a triangle with sides a, b and c:
Semi-perimeter: s = (a + b + c) / 2
Area: A = √( s (s − a) (s − b) (s − c) )
Equivalent form without s: A = ¼ √( (a + b + c)(−a + b + c)(a − b + c)(a + b − c) )
Each factor s − a, s − b and s − c is positive exactly when the triangle inequality holds, which is why the formula only produces a real answer for side lengths that can actually form a triangle.
Worked examples
Sides
s
s(s − a)(s − b)(s − c)
Area
3, 4, 5
6
6 × 3 × 2 × 1 = 36
6
5, 6, 7
9
9 × 4 × 3 × 2 = 216
6√6 ≈ 14.697
7, 8, 9
12
12 × 5 × 4 × 3 = 720
12√5 ≈ 26.833
13, 14, 15
21
21 × 8 × 7 × 6 = 7,056
84
10, 10, 10
15
15 × 5 × 5 × 5 = 1,875
25√3 ≈ 43.301
5, 5, 8
9
9 × 4 × 4 × 1 = 144
12
Triangles with whole-number sides and a whole-number area, like 13-14-15 and 5-5-8, are called Heronian triangles.
Practical tips
Measuring land or a garden bed? Split an irregular shape into triangles with a diagonal, measure the sides of each, and add the Heron areas together.
Use one unit. Convert all three sides to the same unit first; the area comes out in that unit squared.
Keep extra digits. Round only the final area; rounding s or the differences early can noticeably change the result.
Frequently asked questions
What is Heron's formula?
Area = √(s(s − a)(s − b)(s − c)), where a, b and c are the sides and s = (a + b + c)/2 is the semi-perimeter. It gives the area of any triangle from its three sides, without needing a height or an angle.
How do I use Heron's formula step by step?
Add the sides and halve the total to get s. Subtract each side from s. Multiply s by the three differences, then take the square root. For sides 5, 6, 7: s = 9, the differences are 4, 3, 2, the product is 9 × 4 × 3 × 2 = 216 and the area is √216 = 6√6 ≈ 14.70.
Why does the calculator say my sides do not make a triangle?
Any two sides of a triangle must add up to more than the third side (the triangle inequality). With 2, 3 and 6, the two short sides only reach 5, so they cannot meet. If the sum equals the third side exactly, the triangle is flat and its area is 0.
Does Heron's formula work for right and obtuse triangles?
Yes, it works for every triangle: acute, right or obtuse, scalene, isosceles or equilateral. For a 3-4-5 right triangle it gives s = 6 and √(6 × 3 × 2 × 1) = 6, the same as ½ × 3 × 4.
What is the semi-perimeter?
Half of the perimeter: s = (a + b + c)/2. Besides Heron's formula it is used for the inradius, r = Area/s, the radius of the largest circle that fits inside the triangle.
Is Heron's formula accurate for very thin triangles?
Computed naively it can lose precision when one side is tiny compared with the others. A rearranged version that sorts the sides (a ≥ b ≥ c) and uses ¼√((a + (b + c))(c − (a − b))(c + (a − b))(a + (b − c))) stays accurate; this calculator uses it for the decimal answer.
Who was Heron?
Heron (or Hero) of Alexandria was a Greek mathematician and engineer of the first century AD. The formula appears in his work Metrica, although it may have been known earlier, possibly to Archimedes.
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