Heron’s formula calculator finds the area of a triangle from its three side lengths, with no need for the height. Enter sides a, b and c and it shows the semi-perimeter, the area, the perimeter, the three angles and whether the triangle is acute, right or obtuse, along with the steps.
It also checks that the sides can form a triangle. For triangles with two equal sides you can also use the isosceles triangle calculator.
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How Heron’s formula works
The formula
First find the semi-perimeter, which is half the perimeter. Then use it to find the area.
s = (a + b + c) ÷ 2
Area = √(s × (s − a) × (s − b) × (s − c))
Where:
- a, b, c = the lengths of the three sides
- s = the semi-perimeter
The formula is named after Heron of Alexandria.
When three lengths make a triangle
Each side must be shorter than the sum of the other two. If a + b is not greater than c, for example, the sides cannot close up into a triangle and s − c would be zero or negative.
Angles from the three sides
The calculator finds each angle with the law of cosines. For angle A, opposite side a:
cos A = (b² + c² − a²) ÷ (2 × b × c)
Worked examples
Example 1: Sides 7, 8 and 9
s = (7 + 8 + 9) ÷ 2 = 12
Area = √(12 × 5 × 4 × 3) = √720 ≈ 26.8328
The perimeter is 24. The triangle is acute and scalene, with angles of about 48.19, 58.41 and 73.40 degrees.
Example 2: Sides 13, 14 and 15
s = (13 + 14 + 15) ÷ 2 = 21
Area = √(21 × 8 × 7 × 6) = √7056 = 84
The angles are about 53.13, 59.49 and 67.38 degrees, which add up to 180.
Example 3: Sides 3, 4 and 5
s = (3 + 4 + 5) ÷ 2 = 6
Area = √(6 × 3 × 2 × 1) = √36 = 6
Because 3² + 4² = 5², this is a right triangle. The area also equals 3 × 4 ÷ 2 = 6, which confirms the result.
Frequently asked questions
When should I use Heron’s formula?
Use it when you know all three sides but not the height. It avoids finding an angle or a height first.
What is the semi-perimeter?
It is half the perimeter, (a + b + c) ÷ 2. Heron’s formula uses it as the starting value.
Why does the calculator say my sides cannot form a triangle?
One side is as long as, or longer than, the other two added together. For sides 2, 3 and 6, 2 + 3 is less than 6, so no triangle exists.
Does Heron’s formula work for right triangles?
Yes. For a 3, 4, 5 triangle it gives 6, the same as the base times height divided by 2.
Can it find the angles too?
Yes. The calculator uses the law of cosines to give all three angles in degrees.