Workings

Triangle Calculator (Sides, Angles and Area)

Pick what you know — three sides, two sides and the angle between them, or two angles and the side between them — and get everything else.

Triangle Calculator

Your result will appear hereFill in the fields and press “Solve triangle”.

Formula

  • cos A = (b² + c² − a²) ÷ 2bc
  • c² = a² + b² − 2ab·cos C
  • a ÷ sin A = b ÷ sin B = c ÷ sin C
  • area = √(s(s − a)(s − b)(s − c)), s = (a + b + c) ÷ 2
  • Law of cosines for SSS and SAS, law of sines for ASA
  • Heron's formula for the area
  • Angles are in degrees

Three pieces of information, including at least one side, determine a triangle completely.

Worked examples

Sides 5, 6, 7

What do you know?
Three sides (SSS)
Side a
5
Side b
6
Side c
7

Area

14.696938

Side a, b, c: 5, 6, 7

Angle A, B, C: 44.415309°, 57.12165°, 78.463041°

Perimeter: 18

Two sides 5 and 7 with a 60° angle

What do you know?
Two sides and the angle between them (SAS)
Side a
5
Side b
7
Angle C between a and b (degrees)
60 °

Area

15.155445

Side a, b, c: 5, 7, 6.244998

Angle A, B, C: 43.897886°, 76.102114°, 60°

Perimeter: 18.244998

How it works

A triangle is fixed by three pieces of information if at least one is a side. With three sides (SSS) the angles follow from the law of cosines and the area from Heron's formula. With two sides and the angle between them (SAS) the third side comes from the law of cosines first. With two angles and the side between them (ASA) the third angle is 180° minus the other two and the remaining sides come from the law of sines.

The calculator checks that your numbers can form a triangle — each side must be shorter than the sum of the other two, and the angles must add up to less than 180° — and then reports all sides and angles, the perimeter, the area, the inradius and circumradius, and whether the triangle is equilateral, isosceles or scalene and acute, right or obtuse.

For a right triangle where you only need a missing side, the Pythagorean Theorem Calculator is quicker. Shapes with curved edges are covered by the Circle Calculator, and flat and solid shapes more generally by the Area and Volume Calculator.

Assumptions and limits

  • Flat (Euclidean) triangles only. The ambiguous case SSA is not offered because it can have two answers.
  • Right-angle detection allows for tiny rounding differences.
Pythagorean Theorem CalculatorFind the hypotenuse or a missing leg of a right triangle, plus its area, perimeter and angles.Circle CalculatorFind the area, circumference, radius and diameter of a circle from any one of them.Area and Volume CalculatorArea, perimeter, volume and surface area for rectangles, trapezoids, cubes, cylinders, cones and spheres.Square Root CalculatorFind square, cube or nth roots, with simplified radical form and a perfect-power check.Scientific CalculatorType a full expression with brackets, powers, trigonometry, logs and factorials, in degrees or radians.

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