Triangle Calculator

How to use this calculator

1. Fill in any three of the six fields — sides a, b, c and angles A, B, C — including at least one side. The fields sit on the diagram, so each one is beside the part it measures.
2. Choose degrees or radians for the angles. In radians you can type pi/2 or pi/4 as well as a decimal.
3. Calculate to see every remaining side and angle, plus the area, perimeter, heights, medians, inradius, circumradius and vertex coordinates, with the triangle drawn to scale.
4. If your values cannot form a triangle, the calculator says so rather than returning a nonsense result.

The five cases, and why one of them has two answers

Which three values you give decides how the triangle is solved. Four of the five cases have exactly one answer:

• SSS — three sides. Each angle follows from the law of cosines.
• SAS — two sides and the angle between them. The third side follows from the law of cosines.
• ASA — two angles and the side between them. The third angle is 180° minus the other two, then the law of sines gives the sides.
• AAS — two angles and a side that is not between them. Same route as ASA.

The fifth is different. SSA — two sides and an angle that is not between them — can describe two triangles: the side opposite the given angle can swing down to meet the base in two places, giving one shallow triangle and one deep one. This is the ambiguous case, and it is a fact about the geometry rather than a fault in the arithmetic. Where it happens, the calculator shows both triangles instead of quietly choosing one. It can also describe no triangle at all, when that side is too short to reach.

What the calculator works out

Once the triangle is solved, everything else follows: the area by Heron's formula, the perimeter by adding the sides, the height to each side, the median to each side, the inradius and circumradius, and the coordinates of the vertices, centroid, incentre and circumcentre — with vertex A at the origin and B along the x-axis.

The formulas

Each angle comes from the law of cosines, rearranged to solve for the angle opposite a chosen side. For angle A, opposite side a:

cos A = (b² + c² − a²) / (2 × b × c)

The area uses Heron's formula, where s is the semi-perimeter (half the perimeter):

s = (a + b + c) / 2  →  Area = √(s(s − a)(s − b)(s − c))

• a, b, c — the three side lengths
• A — the angle opposite side a (each angle is found the same way, then the third is 180° minus the other two)
• Law of sines — a ÷ sin A = b ÷ sin B = c ÷ sin C, which is what solves the cases that start from an angle
• s — the semi-perimeter, (a + b + c) ÷ 2

When three values form a valid triangle

Not every trio closes into a triangle. The triangle inequality requires each side to be shorter than the sum of the other two — otherwise the two shorter sides cannot reach across the longest one. Sides of 2, 3 and 10 fail this test. Angles must leave room for a third, so any two of them must total less than 180°. And at least one value must be a side: three angles fix a triangle's shape but say nothing about its size, so there would be infinitely many answers. In each case the calculator says what is wrong rather than inventing a number.

A worked example: the 3-4-5 triangle

Enter sides of 3, 4 and 5. For the smallest angle, cos A = (4² + 5² − 3²) / (2 × 4 × 5) = 32 / 40 = 0.8, so A ≈ 36.87°; the same method gives 53.13° and, because they must total 180°, a final angle of exactly 90°. Heron's formula gives s = 6 and an area of √(6 × 3 × 2 × 1) = √36 = 6. Since 3² + 4² = 5², this is the smallest whole-number right triangle — which is why builders use a 3-4-5 to square up corners.

Classifying triangles

The calculator labels every triangle two ways at once:

By sides	By angles
Equilateral — all three sides equal	Right — one 90° angle
Isosceles — two sides equal	Acute — every angle under 90°
Scalene — all three sides different	Obtuse — one angle over 90°

The two labels are independent. Sides of 5, 5 and 8, for example, are isosceles yet obtuse: the angle opposite the long side is cos^-1(−0.28) ≈ 106.26°, and Heron's formula still gives a clean area of 12.

For general trigonometry — sine, cosine and tangent on their own — the scientific calculator handles them directly, and the area calculator covers a triangle from a base and height instead of three sides.