Case 1
- Side a
- 3
- Side b
- 4
- Side c
- 5
6.0000
Open with these valuesHeron's Formula Calculator
6.0000units²
Result: 6.0000 units²Heron's formula gives the area from the three sides alone — no height, no angles. Halve the perimeter to get s, then take √(s(s − a)(s − b)(s − c)). Sides of 3, 4 and 5 give s = 6 and an area of √36 = 6 square units.
The numbers at a glance
Held fixed: Side a 3.000, Side b 4.000.
| Side c | Result |
|---|---|
| 2.000 | 2.9047 |
| 4.000 | 5.5621 |
| 5.000Your value | 6.0000 |
| 6.000 | 5.3327 |
| 8.000 | 0.0000 |
| 10.000 | 0.0000 |
Worked examples
Case 2
- Side a
- 7
- Side b
- 8
- Side c
- 9
26.8328
Open with these valuesCase 3
- Side a
- 13
- Side b
- 14
- Side c
- 15
84.0000
Open with these valuesHow it's calculated
A = √(s(s − a)(s − b)(s − c)), s = (a + b + c) ÷ 2
- StepMeasure the three sides and enter them in one length unit.
- StepCheck the longest side is shorter than the other two combined.
- ResultRead the area in square units; the perimeter is a + b + c.
Reference table
| Sides a, b, c | Exact | Area |
|---|---|---|
| 0.5, 0.5, 0.8 | √0.0144 | 0.1200 |
| 3, 4, 5 | √36 | 6.0000 |
| 5, 5, 5 | 25√3 ÷ 4 | 10.8253 |
| 5, 5, 8 | √144 | 12.0000 |
| 6, 8, 10 | √576 | 24.0000 |
| 7, 8, 9 | 12√5 | 26.8328 |
| 13, 14, 15 | √7056 | 84.0000 |
Questions
What is Heron's formula?
Heron's formula gives the area of a triangle from its three side lengths alone, without the height or any angle. Find the semi-perimeter s = (a + b + c) ÷ 2, then the area is √(s(s − a)(s − b)(s − c)). It is named after Heron of Alexandria, who described it in the first century AD.
How do I calculate a triangle's area from three sides?
Add the three sides and halve the total to get s. Multiply s by (s − a), (s − b) and (s − c), then take the square root of that product. For a 3-4-5 triangle s = 6, so the area is √(6 × 3 × 2 × 1) = √36 = 6 square units.
Why do some sides give an area of zero?
The three lengths must satisfy the triangle inequality: no side may be as long as, or longer than, the other two combined. Sides of 1, 1 and 5 cannot close into a triangle, so they enclose nothing and the area is zero. Check that the longest side is shorter than the sum of the other two.
When should I use Heron instead of ½ × base × height?
Use Heron's formula when you know all three side lengths but not the height — common in surveying, construction and any field triangle where a perpendicular is awkward to measure. If you already have a base and its perpendicular height, ½ × base × height is simpler and just as exact.
Does it work for right and obtuse triangles?
Yes, for every valid triangle: right-angled, acute, obtuse, scalene, isosceles or equilateral. For a right triangle it agrees with ½ × base × height, and the 3-4-5 triangle gives 6 either way.
Sources and last check
Information, not professional advice.