Triangle Area Calculator (3 Sides)

Length of the first side
Length of the second side
Length of the third side
Math

This triangle area calculator finds the area of any triangle when you know the lengths of its three sides (the SSS case). Enter sides a, b, and c and it applies Heron's formula: it first computes the semi-perimeter s = (a + b + c) / 2, then the area A = √[s(s − a)(s − b)(s − c)]. It also returns the perimeter and semi-perimeter, and warns you if the three lengths cannot form a real triangle.

Formula

Heron's Formula: A = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
  • Heron's formula gives the area from the three side lengths alone, with no need to know any angle or height.
  • First find the semi-perimeter: s = (a + b + c) / 2.
  • Then the area is A = √[s(s − a)(s − b)(s − c)].
  • The three sides must satisfy the triangle inequality: the sum of any two sides has to be greater than the third, or no triangle exists.
  • Use the same length unit for all three sides; the area then comes out in those units squared.

Example Calculation

Inputs
  • Side A: 3
  • Side B: 4
  • Side C: 5

Take a 3-4-5 triangle. The semi-perimeter is s = (3 + 4 + 5) / 2 = 6, so the area is √[6(6−3)(6−4)(6−5)] = √[6·3·2·1] = √36 = 6 square units, with a perimeter of 12.

Frequently asked questions

What is Heron's formula?
Heron's formula calculates the area of a triangle directly from the lengths of its three sides: A = √[s(s − a)(s − b)(s − c)], where s is the semi-perimeter (a + b + c) / 2.
When should I use this calculator?
Use it whenever you know all three side lengths of a triangle but not its height or angles, and you want the area, perimeter, or semi-perimeter.
Why do I get an 'invalid triangle' message?
Three lengths only form a triangle if each side is shorter than the sum of the other two. If that triangle inequality fails, no real triangle exists and Heron's formula has no valid result.
What units does the area use?
The area is in the square of whatever length unit you enter. If your sides are in centimetres, the area is in square centimetres.
Do the sides have to form a right triangle?
No. Heron's formula works for any triangle - acute, right, or obtuse - as long as the three sides satisfy the triangle inequality.