Triangle calculator, sides, angles and the drawing.

A triangle calculator solves a triangle from three known values: sides a, b and c, and the angles A, B and C opposite them. Type any three, with at least one side, and it works out the rest: the other sides and angles, the area, perimeter, heights, inradius and circumradius, with a drawing to scale and every step. It handles SSS, SAS, ASA, AAS and the tricky SSA case, and a right triangle mode is a Pythagorean theorem calculator.

How to use the triangle calculator

1. Pick Any triangle or Right triangle.
2. Fill three boxes among Side a, Side b, Side c and Angle A, Angle B, Angle C. In right triangle mode fill two, because angle C is 90 degrees.
3. Choose Degrees or Radians under Angles in. In radians you can type pi/6.
4. The results update as you type. Given values are marked given and the new ones found. The drawing is to scale.
5. Try Try an example for a quick start. Copy result copies the answer as text.

Which case is it

The Pythagorean theorem

In a right triangle, a² + b² = c², where c is the hypotenuse, the side opposite the right angle. With legs 6 and 8: 36 + 64 = 100, so c = 10. With a hypotenuse of 13 and a leg of 5: b² = 169 − 25 = 144, so b = 12. In Right triangle mode you give any two values and the calculator gives the third side, both acute angles, the area and the rest. With a side and an angle it uses sine, cosine and tangent: a hypotenuse of 10 and A = 30° gives a = 10 × sin 30° = 5 and b = 10 × cos 30° = 8.660.

Special right triangles

A 45-45-90 triangle has sides in the ratio 1 : 1 : √2. A 30-60-90 triangle has sides in the ratio 1 : √3 : 2. The 3-4-5 triangle and its multiples, such as 6-8-10, also have whole number sides, as do 5-12-13, 8-15-17 and 7-24-25. The calculator names them when it finds one.

Worked example: SAS

Sides a = 7 and b = 9 with the angle between them C = 50°. The law of cosines gives c² = 49 + 81 − 2 × 7 × 9 × cos 50° = 130 − 80.99 = 49.01, so c = 7.0006. Then sin A = 7 × sin 50° ÷ 7.0006, so A = 49.994° and B = 180° − 50° − 49.994° = 80.006°. The area is ½ × 7 × 9 × sin 50° = 24.130.

The ambiguous case: SSA

Two sides and an angle that is not between them can fit zero, one or two triangles. Take A = 30°, a = 5 and b = 8. Then sin B = 8 × sin 30° ÷ 5 = 0.8, and B can be 53.130° or 180° − 53.130° = 126.870°. Both work, because each leaves room for a third angle: C = 96.870° gives c = 9.928, and C = 23.130° gives c = 3.928. The calculator shows both triangles. If sin B comes out above 1 there is no triangle, and when a is at least as long as b there is exactly one.

Area, heights and circles

• Area comes from Heron's formula, √(s(s − a)(s − b)(s − c)) where s is half the perimeter, or from ½ab sin C.
• Height to a side is 2 × area ÷ that side.
• Inradius is the area ÷ s, the radius of the circle that fits inside. Circumradius is abc ÷ (4 × area), the circle through all three corners.

Common problems

• Sides that cannot close. The two shorter sides must add up to more than the longest. 1, 2 and 10 cannot make a triangle, and the calculator says why.
• Angles that add up to 180 or more. A triangle's angles sum to exactly 180°, so two angles must total less than that.
• Only angles. Three angles give the shape but not the size, so a side is needed.
• Degrees or radians. Check the Angles in box. 1 radian is 57.296°.

For a ratio between a width and a height use the aspect ratio calculator, for roots and powers the scientific calculator, and for converting lengths the unit converter.