Solve a Euclidean plane triangle from sufficient known side lengths and angles, including both valid solutions in an ambiguous SSA case.
Enter known measurements and leave unknown fields blank. The calculator detects SSS, SAS, ASA, AAS, or SSA and reports every valid solution.
Standard notation is used throughout: side a is opposite angle A, side b is opposite B, and side c is opposite C. All sides share one selected length unit.
A + B + C = 180°a + b > c, a + c > b, b + c > aa² = b² + c² − 2bc cos(A)a / sin(A) = b / sin(B) = c / sin(C)a² + b² = c² when C is the right angles = (a + b + c) / 2; area = √[s(s−a)(s−b)(s−c)]P = a + b + cFor a = 3, b = 4, c = 5, the angles are approximately 36.8699°, 53.1301°, and 90°. Area is 6 and perimeter is 12, so it is a scalene right triangle.
For a = b = c = 2, every angle is 60°, area is approximately 1.73205, and perimeter is 6. It is an equilateral acute triangle.
For a = 5, b = 7, and included angle C = 60°, the law of cosines gives c = √39, approximately 6.244998; area is approximately 15.155445.
For A = 30°, a = 10, and b = 12, SSA produces two valid triangles. The calculator reports both rather than discarding the acute- or obtuse-angle solution.
Three angles determine shape but not scale, so AAA cannot determine side lengths, area, or perimeter.
Side a must be opposite angle A, b opposite B, and c opposite C.
Additional measurements are checked against the solved triangle and are not silently ignored.
SSS uses all three side lengths and the law of cosines. Each side must be shorter than the sum of the other two; equality is degenerate and is rejected.
SAS uses two sides and their included angle, then solves the third side with the law of cosines. ASA and AAS derive the third angle from 180° and use the law of sines to scale the missing sides.
SSA supplies an angle, its opposite side, and another side. Since sin(θ) = sin(180° − θ), these measurements can describe zero, one, or two triangles.
Every candidate must have a positive third angle and satisfy the triangle inequality. When two distinct candidates remain, both are displayed.
Once all three sides are known, area is calculated with a numerically stable form of Heron's formula. Perimeter is the side sum and semiperimeter is half that sum.
Equal-side comparisons use numerical tolerance: equilateral has three equal sides, isosceles has two, and scalene has none. Angle classification is acute, right, or obtuse, with tolerance around 90°.
This calculator solves Euclidean plane triangles. It does not handle coordinate, vector, spherical, construction-estimating, roofing, rafter, stair, or three-dimensional geometry.
Provide a sufficient SSS, SAS, ASA, AAS, or SSA set including at least one side.
Usually three independent measurements including at least one side; the exact sufficient combinations depend on their relationships.
They determine shape but not scale, so at least one side is still required.
SSS means all three side lengths are known.
SAS means two sides and their included angle are known.
Both use two known angles and one side; ASA's side lies between the known angles while AAS's does not.
SSA provides two sides and a non-included angle opposite one of those sides.
An angle and its supplement have the same sine, so two distinct triangles can sometimes satisfy the same measurements.
Every pair of sides must have a sum strictly greater than the remaining side.
Depending on the known values, use the law of cosines, law of sines, or the Pythagorean theorem.
Use the law of sines or cosines, or subtract two known angles from 180°.
With three solved sides, this calculator uses a stable form of Heron's formula.
If s is the semiperimeter, area equals the square root of s(s−a)(s−b)(s−c).
The general solver handles it naturally; when the included angle is 90°, the law of cosines reduces to the Pythagorean theorem.
They have respectively no equal sides, two equal sides, or three equal sides.
All angles are below 90°, one is 90°, or one is above 90°, respectively.