The Triangle Theorems Calculator solves a triangle from any three known measurements, sides or angles, by applying whichever classical theorem fits the given combination: the angle sum theorem, the law of sines, or the law of cosines. Enter three values, and the tool identifies the case, names the theorem it used, and returns every remaining side, angle, and derived measurement.
Triangle geometry rests on a small set of theorems that, together, can solve any triangle from three known measurements. Knowing which theorem applies to which combination of knowns is the actual skill; this page names that theorem at every step rather than hiding the method behind a single black-box answer.
Apply the angle sum theorem
Every triangle's three interior angles add up to exactly 180 degrees, regardless of the triangle's shape or size. This is the simplest of the three theorems and the fastest one to apply whenever two angles are already known.
A + B + C = 180°Worked example: a triangle has angles A = 50° and B = 60°. Find angle C.
C = 180 − 50 − 60 = 70°.
Apply the law of sines
The law of sines relates each side to the sine of its opposite angle, useful whenever a side and its opposite angle are both known.
a / sin(A) = b / sin(B) = c / sin(C)This theorem solves the ASA, AAS, and SSA cases directly. ASA and AAS always give a single answer once the angle sum theorem finds the third angle. SSA is the ambiguous case, which can yield zero, one, or two valid triangles depending on the specific values involved.
Apply the law of cosines
The law of cosines generalizes the Pythagorean theorem to any triangle, useful when no angle-side pair is available to start with the law of sines.
c² = a² + b² − 2ab·cos(C)This theorem solves the SSS case (rearranged to solve for each angle) and the SAS case (used directly to find the missing side). Once one side-angle pair is found this way, the law of sines can finish the remaining unknowns more directly.
Identify which theorem applies to your case
| Case | You know | Theorem |
|---|---|---|
| Two angles | any two of the three angles | angle sum theorem |
| ASA / AAS | two angles and a side | law of sines |
| SSA | two sides and a non-included angle | law of sines (ambiguous) |
| SSS | three sides | law of cosines |
| SAS | two sides and the included angle | law of cosines |
Identifying the case correctly before choosing a formula is the step most textbook errors skip, since applying the wrong theorem to a given combination either fails outright or, worse, produces a number that looks plausible but is not the triangle actually described.
Work through a full mixed example
A triangle has sides a = 7, b = 9, and angle C = 40° (the angle between them, an SAS case). 1. Apply the law of cosines to find side c. c² = 7² + 9² − 2(7)(9)cos(40°) = 49 + 81 − 126 × 0.766 ≈ 130 − 96.5 ≈ 33.5.
2. Take the square root. c ≈ √33.5 ≈ 5.79.
3. Apply the law of sines to find angle A. sin(A) = a × sin(C) / c = 7 × sin(40°) / 5.79 ≈ 7 × 0.643 / 5.79 ≈ 0.777, so A ≈ 51.0°.
4. Apply the angle sum theorem to find angle B. B = 180 − 40 − 51.0 = 89.0°.
This sequence, law of cosines first, then law of sines, then the angle sum theorem, is the standard path through an SAS problem, and it shows all three theorems working together on a single triangle.
Understand why three measurements are enough
A triangle has six measurements in total: three sides and three angles. Knowing any three of them, as long as at least one is a side, fixes the entire triangle (with the single exception of the ambiguous SSA case). Three angles alone are not enough, since infinitely many similarly-shaped triangles of different sizes share the same three angles.
Frequently asked questions
What are the main triangle theorems used to solve a triangle?
The angle sum theorem (all angles total 180°), the law of sines (side over sine of opposite angle is constant), and the law of cosines (a generalization of the Pythagorean theorem for any triangle).
How do you know which theorem to use?
Identify which three measurements are given. Two angles alone need only the angle sum theorem. A side-angle pair with another angle or side calls for the law of sines. Three sides, or two sides and the included angle, call for the law of cosines.
Can three angles alone solve a triangle?
No. Three angles fix the triangle's shape but not its size, since infinitely many similar triangles share the same three angles. At least one side length is needed to fix the actual dimensions.
What is the ambiguous case?
SSA (two sides and a non-included angle) can produce zero, one, or two valid triangles from the same three measurements, because the law of sines' inverse sine step can return two different valid angles.
Do these theorems work for any triangle, not just right triangles?
Yes. The law of sines and law of cosines both apply to any triangle. The law of cosines specifically reduces to the Pythagorean theorem as a special case when the included angle is 90°.
What other measurements can be found once a triangle is solved?
Once all six primary measurements (three sides, three angles) are known, area, perimeter, the three heights, the medians, and the radii of the inscribed and circumscribed circles can all be calculated from them.
Is there a shortcut for right triangles specifically?
Yes. A right triangle has one angle fixed at 90°, so the Pythagorean theorem and basic trigonometric ratios solve it more directly than the general law of sines or law of cosines, without needing the full theorem set.
Summary
Solving any triangle comes down to choosing the right theorem for the three known measurements: the angle sum theorem for two known angles, the law of sines for an angle-side pair (ASA, AAS, or the ambiguous SSA), and the law of cosines for SSS or SAS.
A mixed SAS example with sides 7 and 9 and a 40° included angle works through all three theorems in sequence, finding a third side of about 5.79 and the remaining two angles.