Why do some side combinations fail?

The triangle inequality: every side must be shorter than the sum of the other two. 3, 4 and 9 fails because 3 + 4 = 7 < 9, so the two short sides cannot span the long one.

How are the angles calculated?

With the law of cosines, cos A = (b² + c² − a²) / 2bc. The largest angle always faces the longest side, which makes the output easy to sanity-check.

When is the area largest for a fixed perimeter?

The equilateral triangle encloses the most area for a given perimeter, and the more unequal the sides the flatter the triangle becomes.

📐 Scalene Triangle Calculator

Calculate area and angles of a triangle with all different side lengths.

Side a Side b Side c

What is a Scalene Triangle?

A triangle where all three sides have different lengths. All interior angles are also different. Triangle inequality: the sum of any two sides must be greater than the third side.

Heron's Formula (Area)

s = (a + b + c) / 2

S = √(s(s−a)(s−b)(s−c))

A formula for finding the area from only the three side lengths via the semi-perimeter s. Discovered by the mathematician Heron in the 1st century BC.

Law of Cosines (Angles)

cos A = (b² + c² − a²) / (2bc)

cos B = (a² + c² − b²) / (2ac)

cos C = (a² + b² − c²) / (2ab)

Knowing the three side lengths, you can find each interior angle of the triangle.

Triangle Inequality Condition

For sides a, b, c, all three of the following conditions must be satisfied:

a + b > c

a + c > b

b + c > a

Sum of Interior Angles

∠A + ∠B + ∠C = 180°

Worked examples

Sides 3, 4 and 5 give a semi-perimeter of 6, an area of √(6 × 3 × 2 × 1) = 6 and angles of about 36.8699°, 53.1301° and 90°.

Sides 5, 6 and 7 give an area of √216 ≈ 14.6969 and angles of roughly 44.4153°, 57.1217° and 78.4630°, which still add up to 180°.

What the result assumes

Three sides pin down exactly one triangle as long as the longest side is shorter than the sum of the other two; sides such as 3, 4 and 9 leave a gap, so the calculator reports an error. Heron and the law of cosines are well conditioned for ordinary triangles but lose precision in extremely thin ones. Angles are in degrees and always total 180°.

Frequently asked questions

Why do some side combinations fail?: The triangle inequality: every side must be shorter than the sum of the other two. 3, 4 and 9 fails because 3 + 4 = 7 < 9, so the two short sides cannot span the long one.
How are the angles calculated?: With the law of cosines, cos A = (b² + c² − a²) / 2bc. The largest angle always faces the longest side, which makes the output easy to sanity-check.
When is the area largest for a fixed perimeter?: The equilateral triangle encloses the most area for a given perimeter, and the more unequal the sides the flatter the triangle becomes.