Triangle Calculator

Enter three side lengths to get the perimeter, the area by Heron’s formula, whether the triangle is equilateral, isosceles or scalene, whether it is right-angled, acute or obtuse, and its inradius and circumradius. Impossible triangles are politely refused.

How it works: Area comes from Heron's formula using the semi-perimeter (half the perimeter). The angle type is found with Pythagoras: if the square of the longest side equals the sum of the squares of the other two, the triangle is right-angled.

How the Triangle Calculator Works

A triangle is completely determined by any three of its sides and angles, provided the sides can actually form a triangle (the longest side must be shorter than the other two added together). This calculator takes the three side lengths you enter and works out everything else: the perimeter, the area, all three interior angles, whether the triangle is acute, right or obtuse, and which side pattern it follows — equilateral, isosceles or scalene. It also gives the inradius (the radius of the circle that fits exactly inside the triangle, touching all three sides) and the circumradius (the radius of the circle that passes through all three corners).

The area comes from Heron’s formula. First the semi-perimeter is found: s = half the perimeter. Then area = √(s × (s − a) × (s − b) × (s − c)), where a, b and c are the three sides. The angles come from the cosine rule — for the angle opposite side a, cos A = (b² + c² − a²) ÷ (2bc) — and the three angles always add up to 180 degrees, which is a useful sanity check on the answer. The type follows from the sides: all three equal means equilateral; exactly two equal means isosceles; all different means scalene. The angle type follows from the largest angle: under 90 degrees means acute; exactly 90 means right-angled; over 90 means obtuse. The inradius is Area ÷ s, and the circumradius is (a × b × c) ÷ (4 × Area).

How to Use This Calculator

Enter the three side lengths, in the same unit.
Press Calculate.
Read the perimeter and area first — they are the headline results.
Check the angles and the triangle type to confirm the shape matches what you expected.
Use the inradius and circumradius when a circle must fit inside or around the triangle — for example, when planning a rounded cut-out or a circular border.

Example

Example 1. Sides 3, 4 and 5. The perimeter is 12, so the semi-perimeter s is 6. The area is √(6 × 3 × 2 × 1) = √36 = 6. Because all three sides are different, it is scalene, and because 3² + 4² = 25 = 5², the largest angle is exactly 90 degrees — a right-angled triangle. The inradius is 6 ÷ 6 = 1, and the circumradius is (3 × 4 × 5) ÷ (4 × 6) = 60 ÷ 24 = 2.5.

Example 2. Sides 5, 5 and 8. The perimeter is 18, so the semi-perimeter s is 9. The area is √(9 × 4 × 4 × 1) = √144 = 12. Because exactly two sides are equal, it is isosceles, and because 5² + 5² = 50 is less than 8² = 64, the largest angle is more than 90 degrees — an obtuse triangle. The inradius is 12 ÷ 9 ≈ 1.33, and the circumradius is (5 × 5 × 8) ÷ (4 × 12) = 200 ÷ 48 ≈ 4.17.

Understanding Your Result

The most useful check is the angles adding to 180 — if they do not, something was mistyped. The second check is the triangle inequality: if the longest side is equal to or longer than the other two together, no triangle exists, and the calculator will say so rather than inventing an answer. Note that the area and angles depend only on the side lengths, not on how the triangle is drawn or rotated; turning the same triangle on its side changes nothing. And the units follow your inputs: sides in centimetres give area in square centimetres and radii in centimetres — the calculator does not convert units, so keep all three sides in the same unit or the answers will be meaningless.

Triangle Calculator FAQs

What if my sides cannot make a triangle?

The calculator will not produce angles or an area for an impossible set of sides. The rule is simple: the longest side must be shorter than the other two added together. If it is not, check for a typo or a wrong unit on one side.

Do I need to enter the sides in any particular order?

No. The calculator works out which side is longest and which angles sit opposite which sides, so the order you enter them in does not matter.

How are the angles calculated?

From the cosine rule. For the angle opposite side a, cos A = (b² + c² − a²) ÷ (2bc). The other two angles follow the same pattern, and all three always add to 180 degrees.

What is the difference between inradius and circumradius?

The inradius is the radius of the largest circle that fits inside the triangle, touching all three sides. The circumradius is the radius of the circle through all three corners. For a right-angled triangle, the circumradius is exactly half the longest side (the hypotenuse).

Can this calculator find a missing side?

Not directly — it needs all three sides to start from. If you know two sides and the angle between them, or one side and two angles, a different tool (or the cosine and sine rules by hand) will give you the missing side first.