Triangle Calculator
Enter any three values of a triangle, with at least one side, and get the other three, plus the area, the perimeter, the type of triangle and the radii of the inscribed and circumscribed circles. It handles three sides, two sides and the angle between them, two angles and a side, and the ambiguous case of two sides and a different angle, where it shows every triangle that fits. Everything runs in your browser.
Case: SSS
Three values fix a triangle, as long as one of them is a side. With three sides (SSS), the law of cosines gives the angles. With two sides and the angle between them (SAS), the law of cosines gives the third side. With two angles and a side (ASA or AAS), the third angle is 180 minus the other two and the law of sines gives the sides. Two sides and an angle that is not between them (SSA) can have no solution, one, or two, and the tool shows each one. Three angles give only the shape, never the size. The area uses Heron's formula. Everything is worked out in your browser and nothing is uploaded.
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Math & MoneyAbout the Triangle Calculator
A triangle has three sides, a, b and c, and three angles, A, B and C, where each angle is opposite the side with the same letter. Three values fix it, provided one is a side. With three sides (SSS), the law of cosines gives each angle, and the tool checks that the longest side is shorter than the other two added together. With two sides and the angle between them (SAS), the law of cosines gives the third side. With two angles and a side (ASA or AAS), the third angle is 180 minus the other two and the law of sines gives the sides. Two sides and an angle that is not between them (SSA) is the ambiguous case: depending on the numbers, there are no triangles, one, or two, and the tool lists each. For sides 6 and 8 with a 30 degree angle opposite the 6, there are two: one with B = 41.81 degrees and a third side of 11.4, and one with B = 138.19 degrees and a third side of 2.46. Three angles give only the shape, never the size, so a side is required. The area uses Heron's formula, and the tool names the triangle by its sides (equilateral, isosceles or scalene) and by its angles (acute, right or obtuse). Results are shown to ten significant digits. Nothing is uploaded.
How do I solve a triangle with three sides?▾
Use the law of cosines to find each angle: for angle A, cos A = (b squared + c squared - a squared) / (2bc). For sides 3, 4 and 5 that gives angles of 36.87, 53.13 and 90 degrees. The tool does it for you and also gives the area, which is 6.
How do I find the third side from two sides and the angle between them?▾
Use the law of cosines: c squared = a squared + b squared - 2ab cos C. For a = 5, b = 7 and C = 60 degrees, c is about 6.245. If the angle is 90 degrees, it reduces to the Pythagorean theorem.
What is the ambiguous case of a triangle?▾
It happens when you know two sides and an angle that is not between them (SSA). The side opposite the angle may be able to swing to two positions, giving two different triangles, or one, or none. The tool lists every triangle that fits.
Why do three angles not give a triangle?▾
Because every triangle with the same angles has the same shape and any size. 30, 60 and 90 degrees describe a whole family of similar triangles, so you need at least one side to fix the size.
Is my data uploaded?▾
No. The calculation runs in your browser and nothing you enter is sent anywhere.