Example 1
- a = 3 cm
- b = 4 cm
- c = 5 cm
6
What area is 3-4-5? 6 cm². Same as ½·3·4.
Heron's formula calculator
Type three sides in centimetres. Heron’s formula finds the area without a height. With 3, 4 and 5 cm you first get s = 6, then area 6 cm². That is the classic 3-4-5 right triangle.
The sides must satisfy the triangle inequality. 2, 3 and 10 cm do not close a figure and the result is a blank result. The label is cm, the result is cm².
Results
Enter data and click Calculate.
How it works
Heron’s formula finds a triangle’s area from the three sides alone. You do not need a height or an angle. First the semiperimeter s = (a+b+c)/2. Then P = √(s(s-a)(s-b)(s-c)). With 3, 4 and 5 you get s = 6 and √(6·3·2·1) = 6.
Heron of Alexandria wrote that product so each side subtracted from s stays positive. It is another face of ½·a·h: the height is hidden inside the square root. With 5, 5 and 6 cm, s = 8 and the area is 12 cm².
The fields are side a, side b and side c, labelled in cm. The calculator rounds to six decimals. When s is not greater than every side, the triangle does not close and the result stays blank. 2, 3 and 10 cm fail that way.
3-4-5 is right-angled, so ½·3·4 = 6 matches Heron. Same area, two writings. Another triangle without a right angle still passes, as long as the sides close.
The header units switch does not turn typed cm into inches. 3, 4, 5 stay 6 cm². If you have metres, keep one scale in all three fields and remember the result is those units squared.
Equilateral 6, 6, 6: s = 9, area ≈ 15.588457. 5, 12, 13: s = 15, area 30. 7, 8, 9: s = 12, area ≈ 26.832816.
Formula
s = (a + b + c) / 2, then P = √(s(s − a)(s − b)(s − c))
How to use
- Type three sides in cm, for example 3, 4 and 5.
- Click Calculate. The area is 6 cm².
- 5, 5 and 6 give 12 cm².
- 2, 3 and 10 leave a blank field: that is not a triangle.
- Height from ½·a·h lives on another calculator. Here three sides are enough.
3-4-5 gives 6 cm²
Heron’s formula: s = (a+b+c)/2, P = √(s(s-a)(s-b)(s-c)). With 3, 4, 5 cm the area is 6 cm². With 5, 5, 6 cm it is 12 cm².
- Heron
- The three-side area formula. 3-4-5 gives 6 cm². You do not type a height.
- s
- Semiperimeter (a+b+c)/2. For 3, 4, 5 it is 6. It must be larger than each side.
- area
- The result in cm². 6 from 3-4-5. 12 from 5-5-6. 30 from 5-12-13. Blank when the sides do not close.
Examples
Example 2
- a = 5 cm
- b = 5 cm
- c = 6 cm
12
What area with 5, 5 and 6? 12 cm².
Example 3
- a = 5 cm
- b = 12 cm
- c = 13 cm
30
What area is 5-12-13? 30 cm².
Related calculators
Common questions
What area is a 3-4-5 triangle?
6 cm². s = 6, then √(6·3·2·1) = 6. Same as ½·3·4, because the angle between 3 and 4 is right.
What does Heron’s formula mean?
Area from three sides through the semiperimeter s. P = √(s(s-a)(s-b)(s-c)). You do not type a height.
What is s?
The semiperimeter, (a+b+c)/2. For 3, 4, 5 it is 6. For 5, 5, 6 it is 8.
What area with sides 5, 5 and 6?
12 cm². s = 8, √(8·3·3·2) = √144 = 12.
What if the sides do not make a triangle?
The result is a blank result. 2, 3 and 10 cm do not close, because 2+3 is smaller than 10.
Does 5-12-13 work too?
Yes. s = 15, area 30 cm². Another right triangle, ½·5·12 = 30.
Do the units have to be centimetres?
The label says cm and cm². If you type metres, the result is in m², as long as all three sides share one scale.
Does side order change the area?
No. 3, 4, 5 and 5, 3, 4 both give 6. Heron does not ask which side is the base.
Why the square root?
To bring four lengths back to an area, square centimetres. Without the root you would be left with cm⁴.
How is this different from ½·a·h?
Same area. Here you have no h, you have three sides. The height is hidden in s and in the root.
Knowledge sources
The calculator computes the same formula as the definition below.
Page updated in 2026.