Triangle Calculator
Solve any triangle from three known parts, at least one a side: the missing sides and angles, the area, the perimeter and the type, with both answers when two triangles fit.
Give any three of the six parts, sides a, b, c and the angles A, B, C opposite them, with at least one side. Two sides and the angle between them use the law of cosines: with a = 7, b = 9 and C = 60°, c² = 49 + 81 − 63 = 67, so c is about 8.1854, and the other angles come out at 47.78° and 72.22°. Two angles give the third, since all three add up to 180°.
Results assume a flat triangle with every side in the same unit and angles in degrees. Sides, area and perimeter are shown to four decimals and angles to two, so rounding can hide a tiny difference, such as an angle just under 90° that displays as 90.00°.
Fill in exactly three boxes, at least one of them a side, and leave the rest blank. Each angle sits opposite the side with the same letter. Angles are in degrees; sides can use any one unit, up to 1,000,000.
Solved triangle
scalene, acute triangle
- Sides a, b, c
- 7, 9, 8.1854
- Angles A, B, C
- 47.78°, 72.22°, 60.00°
- Area
- 27.2798
- Perimeter
- 24.1854
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About the Triangle Calculator
Three sides fix a triangle only if each is shorter than the other two together, and the angles then follow from the law of cosines, cos A = (b² + c² − a²) / 2bc. One side with two angles uses the angle sum and then the law of sines, which says each side divided by the sine of its opposite angle gives the same number. Two sides with an angle not between them is the tricky case. With a = 6, b = 8 and A = 30°, sin B = 8 × sin 30° / 6, which is two thirds, and two angles have that sine, 41.81° and 138.19°, so two different triangles fit: one with c about 11.4003 and one with c about 2.4561. Shorten side a or change angle A and there may be one triangle or none; with a = 6, b = 8 and A = 60° the sine would have to exceed 1, so no triangle exists. The type names the sides (equilateral, isosceles or scalene) and the largest angle (right at 90°, obtuse above it, acute below), and the area comes from the three sides, as on the area calculator.
Frequently asked questions
How do I find the third side from two sides and an angle?+
If the angle sits between the two sides, use the law of cosines: c² = a² + b² − 2ab cos C. With a = 7, b = 9 and C = 60°, cos 60° is one half, so c² = 49 + 81 − 63 = 67 and c is about 8.1854. If the angle is not between them, the calculator checks for zero, one or two answers.
Why does it show two triangles?+
Two sides and an angle opposite one of them can describe two different shapes, because an acute angle and its supplement have the same sine. Entering a = 6, b = 8 and A = 30° gives B as 41.81° or 138.19°, and both triangles are valid. Your drawing or problem statement decides which one you need.
How do I find a missing angle?+
With two angles known, subtract them from 180°. With three sides known, use the law of cosines for one angle at a time. For the 3, 4, 5 triangle that gives 36.87°, 53.13° and 90.00°.
Can I solve a triangle from three angles?+
No. Three angles give the shape but any size fits, so the calculator asks for at least one side. Enter one side and two angles and it finds the third angle and scales the other sides by the law of sines.
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