How to solve a triangle
- Enter any three known values, at least one of them a side: three sides (SSS), two sides and the angle between them (SAS), two angles and a side (ASA or AAS), or two sides and an angle not between them (SSA).
- The calculator finds the missing sides and angles, the area, perimeter, heights and radii, and draws the triangle to scale.
- For a right triangle, press Right triangle to set angle C to 90° and enter two more values, such as the legs a and b.
The laws behind it
Sides are labelled a, b and c, and each angle (A, B, C) is opposite the side with the same letter. With three sides, the law of cosines gives each angle: cos A = (b² + c² − a²) ÷ 2bc. With two angles, the third is 180° minus the other two, and the law of sines (a ÷ sin A = b ÷ sin B = c ÷ sin C) gives the remaining sides. With two sides and the included angle, the law of cosines gives the third side first. The area uses ½ · b · c · sin A, which equals Heron's formula.
The SSA case is famous for being ambiguous: two sides and a non-included angle can describe no triangle, one triangle or two different triangles. When two exist, both solutions are listed. The classic 3-4-5 example is a right triangle with area 6. Values are calculated in double precision and shown to four decimal places; units are whatever you use for the sides. Everything runs in your browser.
Can I solve a triangle from three angles?
No. Three angles fix only the shape; you need at least one side for the size.
How do I use it for a right triangle?
Set C to 90° and enter two other values; this also works as a Pythagorean theorem calculator.
Why do I get two solutions?
With two sides and an angle not between them (SSA), the third vertex can sometimes be in two places.