Triangle Calculator — area from three sides (Heron's formula)

How it works

Heron's formula finds the area of any triangle from just its three sides — no height or angle needed. First compute the semiperimeter, half the perimeter:

s = (a + b + c) ÷ 2

Then the area is the square root of s times its differences from each side:

A = √(s(s − a)(s − b)(s − c))

For the three lengths to form a real triangle at all, they must satisfy the triangle inequality: each side must be shorter than the sum of the other two. If not (say 1, 1, 5), no triangle exists and there's no area to compute.

Worked example

Take the 3-4-5 triangle. The semiperimeter is s = (3 + 4 + 5) ÷ 2 = 6. Then:

A = √(6 × (6 − 3) × (6 − 4) × (6 − 5)) = √(6 × 3 × 2 × 1) = √36 = 6

Its perimeter is 3 + 4 + 5 = 12. Since 3-4-5 is a right triangle, you can check this against ½ × base × height = ½ × 3 × 4 = 6 — they agree. An equilateral triangle of side 5 has area 25√3 ⁄ 4 ≈ 10.83.

Frequently asked questions

How do I find a triangle's area from its sides?

Use Heron's formula. Add the three sides and halve to get the semiperimeter s, then take A = √(s(s − a)(s − b)(s − c)). For sides 7, 8, 9: s = 12, so A = √(12 × 5 × 4 × 3) = √720 ≈ 26.83. It works for any triangle whose sides are known.

What is the semiperimeter?

Half the perimeter — s = (a + b + c) ÷ 2. It's the quantity Heron's formula is built around: the area comes from s and how far it exceeds each side. For a 3-4-5 triangle the perimeter is 12 and the semiperimeter is 6.

Why do the three sides have to satisfy the triangle inequality?

Because a side longer than the sum of the other two can't close into a triangle — the two shorter sides won't reach across the gap. So a, b and c must each be less than the sum of the other two (1, 1, 5 fails). When they don't, Heron's formula would take the square root of a negative number, which signals there's no triangle.

Do I need the height or any angles?

No — that's the appeal of Heron's formula. If you only know the three side lengths it gives the area directly, without first finding a height or angle. If you happen to know a base and its perpendicular height instead, ½ × base × height is simpler.

Does it work for right, isosceles and equilateral triangles?

Yes, for every triangle. A right triangle's Heron area matches ½ × leg × leg; an equilateral triangle of side a gives A = √3 ⁄ 4 × a². Heron's formula doesn't need to know which type it is — the sides carry all the information.

How do I find the height from the area?

Rearrange area = ½ × base × height: the height to a chosen base is 2 × area ÷ base. So a triangle of area 6 on a base of 4 has height 2 × 6 ÷ 4 = 3. Each side can serve as a base, giving a different corresponding height.