Triangle Calculator
Solve general triangles from three known measurements: get every missing side and angle, the area and the perimeter, or the reason no triangle fits.
How the labels work
The three corners are A, B and C, and each side is named after the corner across from it: side a is opposite angle A, b is opposite B, and c is opposite C. Keeping these pairs straight matters most in the SSA case, where the known angle has to sit across from one of the known sides.
How missing sides, angles and area are calculated
Three sides give the angles through a half-angle form of the Law of Cosines that stays accurate for very thin triangles, and the area through Heron's formula in Kahan's stable arrangement. Two sides and the angle between them give the third side through the Law of Cosines. With two angles, the third is 180° minus the other two, and the Law of Sines scales the remaining sides. Area otherwise comes from ½ × side × side × sine of the angle between them.
If your triangle has a right angle and you know two sides, the Pythagorean Theorem Calculator shows the a² + b² = c² steps.
The SSA (ambiguous) case: zero, one or two triangles
Suppose you know a, b and angle A, with A acute. The height the side a has to reach is h = b sin A. If a is shorter than h there is no triangle; if a equals h there is exactly one, with a right angle at B; if a is between h and b there are two; and if a is at least b there is one. When A is 90° or more, a has to be the longest side. The calculator applies the same test to whichever side and angle you enter, so the letters can be in any position.
Why there may be no triangle
Each side must be shorter than the other two combined. Sides of 1, 2 and 3 lie on a straight line, and 1, 2 and 3.5 can't meet at all. When a check fails, the result shows the numbers involved instead of a generic error, and your entries stay in place so you can change one value.
Units and precision
Side lengths have no built-in unit. If you enter inches, the sides and perimeter are in inches and the area is in square inches; just use the same unit for every side. Sides can be from 0.0001 to 1,000,000, with the longest at most 1,000,000 times the shortest, and angles must be greater than 0.01° and less than 179.99°. Every candidate is checked against your entries, the 180° angle sum and the triangle inequality before it's rounded to at most six decimal places. A triangle so close to flat that this precision can't tell it apart from a line is reported as unstable rather than shown with a false zero area.
Privacy
Everything is calculated in this browser tab. Your measurements aren't saved by this page, added to the page address or sent in analytics; a copy or download happens only when you choose it. See our privacy policy and disclaimer.
Frequently asked questions
What three values are enough?
Any three of the six measurements can define a triangle as long as at least one is a side: three sides (SSS), two sides and the angle between them (SAS), two angles and the side between them (ASA), two angles and a side that is not between them (AAS), or two sides and an angle opposite one of them (SSA). Three angles alone fix the shape but not the size, so they can't give side lengths.
Why are there two SSA solutions?
When you know two sides and an angle opposite one of them, the other side can sometimes swing to two different positions and still close the triangle. For a = 7, b = 10 and A = 30°, angle B can be about 45.585° or about 134.415°, giving two different triangles. The calculator shows both together, before the step-by-step working, instead of picking one.
Why is there no triangle?
Some combinations can't close. Each side must be shorter than the other two combined, two known angles must add up to less than 180°, and in the SSA case the side opposite the known angle must reach at least the height h; if that angle is 90° or more, that side must also be longer than the other known side. The calculator shows the numbers that fail, such as h = 10 sin(30°) = 5 when a = 4 is too short.
When should I use the Pythagorean tool?
If you have a right triangle and know two of its sides, the Pythagorean Theorem Calculator finds the missing side with the a² + b² = c² steps. Use this triangle calculator for any triangle, including ones without a right angle, or when some of your known values are angles.
Why does rounding change the displayed angle sum?
The calculator checks the unrounded values, where the angles add up to 180°. Each angle is then rounded to at most six decimal places for display, so adding the displayed numbers can differ from 180 in the last digit. Values marked ≈ are rounded.
Sources and method
- OpenStax Precalculus 2e, Chapter 8 key equations (Law of Sines, Law of Cosines, area formulas, Heron's formula), reviewed September 25, 2026
- OpenStax Precalculus 2e, Chapter 8 key concepts (the ambiguous SSA case), reviewed September 25, 2026
- W. Kahan, Miscalculating Area and Angles of a Needle-like Triangle (numerically stable area and angles), reviewed September 25, 2026