Use the free Triangle Calculator below: your data never leaves your browser — all work happens on your device, and nothing is uploaded to any server.
What a Triangle Calculator Solves
Every triangle has six measurements: three sides and three angles. If you know any three of them, with at least one side, the rest can be worked out. This tool handles the two most common cases: three sides (SSS) and two sides with the angle between them (SAS). From those inputs it finds the missing sides and angles, plus the perimeter and the area, and it checks that the three angles add up to 180 degrees.
The mathematics behind it is three classic results. The law of cosines finds a side from two sides and their included angle: c² = a² + b² − 2ab·cos(C). The law of sines relates sides to the sines of their opposite angles: a/sin(A) = b/sin(B) = c/sin(C). And Heron’s formula finds the area from the three sides alone, using the semi-perimeter s = (a + b + c)/2: area = √(s(s−a)(s−b)(s−c)). The tool applies them in the right order so you do not have to.
How to Use This Calculator
Choose the mode that matches what you know. In SSS mode, enter the three side lengths. In SAS mode, enter two sides and the angle between them in degrees. Press Solve triangle. The tool prints all three sides, all three angles, the perimeter and the area, plus an angle-sum check that should read 180 degrees.
The tool also refuses impossible triangles. In SSS mode, the two shorter sides must add up to more than the longest side; otherwise no triangle can close, and the tool says so instead of printing angles for a shape that cannot exist. In SAS mode, the angle must be between 0 and 180 degrees. These guards catch the two most common input slips: a mistyped side and an angle entered in radians by accident.
A Worked Example: Three Sides
Take sides 5, 6 and 7. The perimeter is 18. The semi-perimeter is 9, so Heron’s formula gives area = √(9 × 4 × 3 × 2) = √216 ≈ 14.6969. The angle opposite side 5 comes from the law of cosines: cos(A) = (6² + 7² − 5²)/(2 × 6 × 7) = (36 + 49 − 25)/84 = 60/84, so A ≈ 44.42 degrees. The other angles follow the same way, and all three add to 180 degrees.
Try it in the tool and compare each number. The value of seeing the full set at once is that a single wrong input shows up everywhere: if one angle looks strange, the area usually looks strange too, and that double signal tells you to re-check your inputs rather than trust one number.
A Worked Example: Two Sides and an Angle
Take sides 8 and 5 with an included angle of 60 degrees. The law of cosines gives the third side: c² = 8² + 5² − 2 × 8 × 5 × cos(60°) = 64 + 25 − 40 = 49, so c = 7. The area is simpler: area = ½ × 8 × 5 × sin(60°) ≈ 17.3205. The remaining angles come from the law of sines, and they complete the triangle.
This SAS case is the one surveyors and builders meet most: two measured distances and the angle between them, with the third distance needed. It is also the case where a quick sketch helps most, because drawing the two sides at the given angle shows at once whether the answer is plausible.
Common Mistakes
The classic mistake is mixing up which angle is “included”. In SAS mode, the angle must sit between the two sides you entered. If you enter the wrong angle, the law of cosines still produces a number, but it answers a different triangle from the one you meant. Sketch the triangle and mark the angle before you type.
Another mistake is entering angles in radians when the tool expects degrees. A 60-degree angle typed as 1.047 (its radian value) would be treated as a 1-degree angle, and the triangle would come out long and thin. If your triangle looks nothing like your sketch, check the units first.
A third mistake is forgetting the triangle inequality in SSS mode: three numbers are not automatically a triangle. Sides 2, 3 and 10 cannot close, no matter how the arithmetic is arranged. The tool refuses them, which is the correct behaviour, not a bug.
Try It Yourself
Start with the 3-4-5 right triangle in SSS mode and confirm the right angle reads 90 degrees. Then try 5-5-5 and confirm all angles read 60 degrees. Then switch to SAS mode with sides 8 and 5 and angle 60, and confirm the third side is 7. Those three checks cover the main paths through the tool. Use it as often as you like; every calculation runs in your browser, and nothing you type is sent anywhere.