How this calculator works
Enter whichever three measurements you know about a triangle—three sides (SSS), two sides and the angle between them (SAS), two angles and the side between them (ASA), or two sides and an angle that is not between them (SSA)—and this calculator solves for everything else: the remaining sides, all three angles, the perimeter, the area, and whether the triangle is right/acute/obtuse and equilateral/isosceles/scalene.
SSA deserves special attention because it is the one case where the given measurements do not always determine a unique triangle. Depending on how the given side and angle relate, there can be zero, one, or two valid triangles—this is called the ambiguous case, and this calculator detects which situation applies and reports it plainly rather than silently picking one answer.
The formulas
Law of cosines (SSS, SAS): c^2 = a^2 + b^2 - 2ab*cos(C)Law of sines (ASA, SSA): a/sin(A) = b/sin(B) = c/sin(C)Area (SSS, Heron's formula): s = (a+b+c)/2, Area = sqrt(s(s-a)(s-b)(s-c))Area (two sides + included angle): Area = (1/2)*a*b*sin(C)Triangle inequality: each side must be shorter than the sum of the other two, or no triangle exists. Angle sum: all three angles always add up to 180 degrees.
Worked example: SSA ambiguous case, a=7, b=10, angle A=40 degrees
- The 'swing height' is h = b*sin(A) = 10*sin(40 deg) = 6.428.
- Since h (6.428) < a (7) < b (10), this configuration has two valid solutions.
- Solving the law of sines gives two possible values for angle B: about 66.98 degrees (acute) and its supplement, about 113.02 degrees (obtuse).
- Each choice of B gives a different angle C and side c, so two distinct triangles both satisfy a=7, b=10, A=40 degrees—both are shown as separate solutions.
Frequently asked questions
What is the SSA ambiguous case, and why does it happen?
SSA means you know two sides and an angle that is opposite one of them, not the angle between them (that would be SAS instead). Because the law of sines can return two possible angles (an acute one and its supplement) that both fit the given information, the same two sides and angle can sometimes be swung into two different valid triangles, one triangle, or none at all.
How do I know if my SSA input has zero, one, or two solutions?
Compare side a (opposite the given angle) to the 'swing height' h = b*sin(A). If a is less than h, no triangle exists. If a equals h, there's exactly one (a right triangle). If h is less than a and a is less than b, there are two. If a is at least as large as b, there's exactly one.
Why do I need the triangle inequality for SSS?
Three lengths only form a triangle if each one is shorter than the sum of the other two—otherwise the two shorter sides can't reach far enough to meet and close the shape. For example, sides 1, 2, and 10 fail because 1+2 is far less than 10.
How is a triangle classified as right, acute, or obtuse?
Compare the square of the longest side to the sum of squares of the other two (the Pythagorean relationship). If they're equal, it's right; if the longest side's square is smaller, it's acute; if larger, it's obtuse.
What's the difference between area formulas used here?
When all three sides are known (SSS), Heron's formula computes area directly from the side lengths. When two sides and the included angle are known (SAS), the simpler (1/2)*a*b*sin(C) formula avoids needing the third side first.