Triangle Solver Guide: Area, Angles & Side Formulas for Every Triangle Type
With any three triangle measurements (sides and/or angles), you can find all remaining values. Learn the Law of Sines, Law of Cosines, and Heron's formula.
Right Triangles: Pythagorean Theorem & Trig Ratios
For right triangles (containing a 90° angle), the Pythagorean Theorem c² = a² + b² relates the three sides, where c is the hypotenuse (longest side, opposite the right angle). Trigonometric ratios provide side-to-angle relationships. SOH-CAH-TOA: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent. Worked example:
Law of Sines (AAS, ASA, SSA Cases)
The Law of Sines states that in any triangle, each side divided by the sine of its opposite angle gives the same ratio. Use it when you know: two angles and any side (AAS or ASA), or two sides and the angle opposite one of them (SSA — watch for the ambiguous case). Worked example: Triangle with A = 35°, B = 85°, b = 12. Find side a. a/sin(35°) = 12
Law of Cosines (SAS, SSS Cases)
The Law of Cosines generalizes the Pythagorean Theorem for non-right triangles. Use it when you know: two sides and the included angle (SAS), or all three sides (SSS — to find angles). Worked example (SAS): a = 7, b = 10, C = 45°. Find side c. c² = 49 + 100 − 2(7)(10)cos(45°) = 149 − 140(0.7071) = 149 − 98.99 = 50.01. c ≈ 7.07. Worked example (SSS
Area Formulas
Multiple formulas compute triangle area depending on available information. The base-height formula requires a base and its perpendicular height. When two sides and an included angle are known, use the trigonometric area formula. When all three sides are known, Heron's formula applies. Heron's formula worked example: a = 6, b = 8, c = 10. s = (6+8+
Frequently Asked Questions
When do I use Law of Sines vs Law of Cosines?
Law of Sines: use when you have AAS, ASA, or SSA (two angles + any side, or two sides + angle not between them). Law of Cosines: use when you have SAS (two sides + included angle) or SSS (all three sides). For right triangles, the Pythagorean Theorem and SOH-CAH-TOA are always si
What is the ambiguous case in Law of Sines?
The ambiguous case (SSA) occurs when you know two sides and an angle opposite one of them. Depending on the values, there may be 0, 1, or 2 valid triangles. Check: compute sin(B) = b·sin(A)/a. If sin(B) > 1: no triangle possible. If sin(B) = 1: exactly one right triangle. If sin(
How do I find the area of a triangle with three sides?
Use Heron's formula: (1) Calculate the semi-perimeter s = (a+b+c)/2. (2) Area = √(s(s−a)(s−b)(s−c)). Example: a=9, b=12, c=15. s=18. Area = √(18×9×6×3) = √2916 = 54. This is a 9-12-15 right triangle (a 3-4-5 triple scaled by 3), verifiable as ½×9×12 = 54 ✓.
What is the difference between similar and congruent triangles?
Congruent triangles have identical side lengths and angles — they are the same size and shape. Congruence tests: SSS, SAS, ASA, AAS, HL (right triangles only). Similar triangles have the same angles but different sizes — their sides are proportional. Similarity tests: AA (two equ
How is the Pythagorean Theorem proved?
Over 370 distinct proofs exist, more than for any other theorem. The simplest visual proof: arrange four identical right triangles inside a square of side (a+b). The area of the large square = (a+b)² = a² + 2ab + b². The four triangles occupy area 4 × ½ab = 2ab. The remaining inn