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Triangle Calculator

Three lengths either close into a triangle or they do not, and knowing which comes first: trusses, roof braces and plots depend on sides that meet. When they do, Heron's formula turns three lengths into an area with no height to measure, and the sides reveal the triangle's character — equilateral, right-angled, isosceles or scalene. This calculator does both, and refuses politely when the sides cannot close.

Enter values

Each side must be shorter than the other two added up

Areas come back in that unit squared — cm gives cm²

Results update as you type.

Results

Area (Heron's formula)
6 cm²
Perimeter
12 cm
Triangle type
Right-angled

How is the Triangle calculated?

Three number fields take the side lengths, and a menu picks the unit — centimetres, metres or feet. The calculator first checks the triangle inequality: every side must be strictly shorter than the other two added together, otherwise three straws lie flat and enclose nothing. Valid sides get a semi-perimeter, Heron's square-root area, the perimeter and a name from the sides: all equal is equilateral, exact Pythagoras means right-angled, two equal is isosceles, anything else scalene.

Formula

The semi-perimeter is s = (a + b + c) ÷ 2 and Heron's formula gives the area as √(s × (s − a) × (s − b) × (s − c)), where a, b and c are your three sides. Each side must be shorter than the other two summed. Sorting the sides also classifies the shape: a² + b² = c² makes it right-angled, and repeated lengths make it equilateral or isosceles.

Example

Sides 3, 4 and 5 cm: the semi-perimeter is (3 + 4 + 5) ÷ 2 = 6, and Heron's formula gives √(6 × 3 × 2 × 1) = √36 = 6 cm², with a 12 cm perimeter. Since 3² + 4² = 9 + 16 = 25 = 5², it is right-angled — the classic builder's triangle. Sides 5, 5 and 6 give s = 8 and area √(8 × 3 × 3 × 2) = √144 = 12 cm², an isosceles triangle. Sides 1, 2 and 10 fail immediately: 1 + 2 is well short of 10.

Things to remember

Heron's formula earns its keep whenever the height is awkward to measure — three tape measurements beat one slanting height. The inequality check is just as useful: it catches impossible orders before the saw comes out. Right-angled triangles are quicker in the Pythagorean calculator, which links back.

Frequently Asked Questions

How do you find a triangle's area from three sides?

Use Heron's formula: halve the perimeter to get s, then take √(s(s − a)(s − b)(s − c)). For sides 3, 4 and 5 the semi-perimeter is 6 and the area is √(6×3×2×1) = 6. No height or angle is needed, which is why surveyors favour it.

What makes three lengths a valid triangle?

Each side must be shorter than the other two added together. 3, 4 and 5 pass because 3 + 4 exceeds 5, while 1, 2 and 10 fail because 1 + 2 falls far short of 10 — such lengths lie flat on a line and enclose no area, so the calculator reports them as invalid.

How does the calculator name the triangle type?

From the sides alone. Three equal lengths mean equilateral; exactly two equal lengths mean isosceles; and when the sorted sides satisfy a² + b² = c² the triangle is right-angled, checked before the isosceles label because the corner matters more. Anything else is scalene.

What is the semi-perimeter and why does it matter?

It is half the perimeter, written s = (a + b + c) ÷ 2, and it is Heron's formula's scaffolding. Sides 5, 5 and 6 give s = 8, so the area becomes √(8 × 3 × 3 × 2) = 12 — the formula simply measures what remains of s after each side is subtracted.

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