Heron’s formula
Heron’s formula gives the area of any triangle from its three side lengths a, b, and c: A = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) / 2 is the semiperimeter. You do not need a height or any angles.
| Use it to | Formula |
|---|---|
| Semiperimeter | s = (a + b + c) / 2 |
| Area | A = √[s(s − a)(s − b)(s − c)] |
| Without s | A = ¼√[(a + b + c)(−a + b + c)(a − b + c)(a + b − c)] |
What each letter means
- a, b, c
- The three side lengths, all in the same unit
- s
- The semiperimeter: half of the perimeter
- A
- The area, in square units
Heron’s formula calculator
Enter three sides. The steps show the semiperimeter and Heron’s formula, along with every angle.
What do you know?
Each side must be shorter than the other two combined.
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The idea
The letter s is the semiperimeter, or half the distance around the triangle: s = (a + b + c) / 2. Heron’s formula works for any valid triangle when all three sides are known.
When to use it
- You know all three side lengths but not a height or angle.
- Measuring land, roofs, or fabric, where sides are easier to measure than heights.
- Checking an area after solving a triangle with the law of cosines.
- If a perpendicular height is already known, A = ½bh is quicker.
Worked examples
1Sides 5, 6, and 7
- s = (5 + 6 + 7) / 2 = 9.
- s − a = 4, s − b = 3, s − c = 2.
- A = √(9 × 4 × 3 × 2) = √216.
- √216 = 6√6 ≈ 14.697.
Answer: The area is about 14.697 square units.
2Sides 13, 14, and 15
- s = (13 + 14 + 15) / 2 = 21.
- s − a = 8, s − b = 7, s − c = 6.
- A = √(21 × 8 × 7 × 6) = √7056 = 84.
Answer: The area is exactly 84 square units.
3When the sides do not make a triangle
- Sides 2, 3, and 6 give s = 5.5.
- s − c = 5.5 − 6 = −0.5, so the product under the square root is negative.
- This happens because 2 + 3 is not greater than 6.
Answer: No triangle exists, so there is no area to find.
Where Heron’s formula comes from
- Start from A = ½ab sin C, so 16A² = 4a²b² sin² C = 4a²b² − (2ab cos C)².
- The law of cosines gives 2ab cos C = a² + b² − c², so 16A² = 4a²b² − (a² + b² − c²)².
- Factor this difference of squares twice: 16A² = (c² − (a − b)²)((a + b)² − c²).
- That equals (−a + b + c)(a − b + c)(a + b − c)(a + b + c) = 2(s − a) × 2(s − b) × 2(s − c) × 2s.
- Divide by 16 and take the square root: A = √[s(s − a)(s − b)(s − c)].
Common mistakes
- Using the full perimeter instead of the semiperimeter s.
- Forgetting the square root in the final step.
- Mixing units. Use one unit for all sides; the area is in that unit squared.
- If s − a, s − b, or s − c is zero or negative, the sides cannot form a triangle.
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Keep exploring
Heron’s formula: common questions
Wondering about the why? Start here.
What is s in Heron’s formula?
The letter s is the semiperimeter: half of the triangle’s perimeter, s = (a + b + c)/2. For sides 3, 4, and 5, s = 6. It is not a side length, and using the full perimeter instead of s gives the wrong area.
Calculate s and the areaWhen should I use Heron’s formula?
Use it when you know all three side lengths but not a height or an angle. If you already know a base and its perpendicular height, A = bh/2 is quicker. If you know two sides and the angle between them, use A = ½ab sin C.
Compare area formulasDoes Heron’s formula work for right and obtuse triangles?
Yes. It works for every valid triangle. For a 3-4-5 right triangle it gives √(6 × 3 × 2 × 1) = 6, which matches ½ × 3 × 4. The sides only need to satisfy the triangle inequality.
Review right trianglesWho discovered Heron’s formula?
It is named after Heron (or Hero) of Alexandria, who recorded the formula with a proof in his Metrica around the first century AD. Some historians think Archimedes knew it earlier.