Enter any three elements — sides and angles — and solve the whole triangle with the Law of Sines and Cosines. Get every side and angle, area, perimeter and both radii, with a diagram.
Area
6
Perimeter
12
Calculator
Sides
a=3, b=4, c=5
Angles
A=36.87°, B=53.13°, C=90°
Area
6
Perimeter
12
Circumradius (R)
2.5
Inradius (r)
1
How a triangle is solved
A triangle is fully determined by three elements as long as at least one is a side. The Law of Cosines (c² = a² + b² − 2ab·cosC) handles three sides (SSS) or two sides and the included angle (SAS); the Law of Sines (a/sinA = b/sinB = c/sinC) handles two angles and a side (ASA/AAS).
The SSA case — two sides and a non-included angle — can be ambiguous: zero, one or two triangles may satisfy it. The solver checks each candidate and returns all valid triangles. Area is found by Heron’s formula, the circumradius from R = abc/4·Area and the inradius from r = Area/s.
Why do I sometimes get two triangles?
When you give two sides and an angle opposite one of them (the SSA case), the unknown angle has two possible values that both close the triangle. This is the classic ambiguous case, and both solutions are shown.
What counts as a valid set of inputs?
Exactly three of the six elements (three sides and three angles), with at least one side. Three angles alone only fix the shape, not the size, so a side is always required.
Are angles in degrees or radians?
Angles are entered and reported in degrees. The three interior angles always add up to 180°.
Results are estimates. Verify with a professional for important decisions.
About this calculator
This calculator solves any triangle when you supply exactly three known elements — sides and angles in any combination (SSS, SAS, ASA, AAS, or the ambiguous SSA case). Enter what you know and the tool fills in all six elements along with area, perimeter, circumradius, and inradius.
How to read your results
After you type three known values the result card appears on the right. It shows a scaled diagram of your triangle with vertices labelled A, B, C, followed by a data list: all three sides, all three angles in degrees, area, perimeter, circumradius, and inradius. If the SSA input is ambiguous and produces two valid triangles, both are shown as separate cards.
Worked example
Enter the three sides of a right triangle: a = 3, b = 4, c = 5.
The solver returns angle A approximately 36.87 degrees, angle B approximately 53.13 degrees, and angle C exactly 90 degrees. Area = 6, perimeter = 12, circumradius = 2.5, inradius = 1.
Frequently asked questions
What is the SSA ambiguous case?
When you know two sides and the angle opposite one of them (SSA), the given data may fit zero, one, or two distinct triangles. The calculator checks all possibilities and returns every valid solution, so you see immediately whether the problem is ambiguous.
Do angles need to be in degrees or radians?
All angles are entered and displayed in degrees. The underlying law-of-sines and law-of-cosines calculations convert to radians internally, so you never need to do that conversion yourself.
How accurate are the results?
The calculator uses JavaScript double-precision arithmetic, which gives about 15 significant digits. Results are displayed to three decimal places, which is more than sufficient for geometric construction, drafting, or coursework.
How it's calculated
For SSS, angles are found with the law of cosines: A = arccos((b^2 + c^2 - a^2) / 2bc), then the third angle by subtraction. For SAS, the unknown side is found with c^2 = a^2 + b^2 - 2ab cos C, and angles follow from the law of cosines. For ASA/AAS, the third angle is 180 degrees minus the sum of the two known angles, then sides are found via the law of sines ratio a/sin A = b/sin B = c/sin C. SSA applies the law of sines to find the candidate angle(s) for the second known side, then checks whether the third angle is positive. Area uses Heron's formula: sqrt(s(s-a)(s-b)(s-c)) where s is the semi-perimeter. Circumradius = abc / (4 * area); inradius = area / s.
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