Triangle calculator

Give three facts about a triangle and get all of its sides, angles and area.

Formula Geometry

Area

Perimeter: 12
Side a: 3
Side b: 4
Side c: 5
Angle A: ≈ 36.8699 °
Angle B: ≈ 53.1301 °
Angle C: 90 °

Right scalene triangle.

How it works

Side a is opposite angle A, b opposite B and c opposite C. With three sides, the law of cosines gives each angle. With two sides and the angle between them, it gives the third side.

With two angles, the third is 180° minus the other two, and the law of sines gives the remaining sides. The area uses Heron’s formula in a form that stays accurate for very flat triangles.

Formula

c² = a² + b² − 2ab cos C
a ÷ sin A = b ÷ sin B = c ÷ sin C
A + B + C = 180°
area = ½ab sin C = √(s(s − a)(s − b)(s − c)), s = (a + b + c) ÷ 2

Worked example

Sides 3, 4 and 5 give angles of about 36.87°, 53.13° and exactly 90°. The perimeter is 12 and the area is ½ × 3 × 4 = 6.

Questions people ask

Why isn’t SSA (two sides and a non-included angle) offered?

SSA can fit zero, one or two different triangles, so it does not always give a single answer.

Do the angles have to add up to 180°?

Yes, in any flat (Euclidean) triangle the three angles always total 180°.

What is the triangle inequality?

Any two sides together must be longer than the third. 2, 3 and 6 can’t form a triangle because 2 + 3 < 6.