OneTaskBy Tim King · Updated
Triangle Calculator
SSS, SAS and ASA Triangle Solver for Sides, Angles, Area and Type
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How the Triangle Calculator works
This triangle calculator solves a whole triangle from any three parts that pin it down: three sides, two sides and the angle between them, two angles and a side, or two sides and an angle that is not between them. It returns every side and angle, the area, perimeter, heights and type, and writes out each step of the working.
A triangle has six parts, three sides and three angles, and knowing three of them (as long as one is a side) is enough to find the rest. The labels follow the usual convention: side a sits opposite angle A, side b opposite angle B and side c opposite angle C. The three angles always add up to 180°.
How the triangle calculator solves each case
The calculator uses two formula rules. The law of cosines, c² = a² + b² − 2ab cos C, handles the SSS and SAS cases. The law of sines, a ÷ sin A = b ÷ sin B = c ÷ sin C, handles ASA, AAS and SSA. Area comes from Heron's formula once all three sides are known.
The SSA case is the tricky one. Two sides and an angle that is not between them can fit no triangle, one triangle or two different triangles. When two fit, the calculator shows both, labelled Triangle 1 and Triangle 2.
First example. Sides 7, 8 and 9 give angles of 48.19°, 58.41° and 73.40°, an area of 26.83 and a perimeter of 24. Every angle is under 90°, so it is an acute scalene triangle.
Second example. Sides 5 and 7 with 60° between them give a third side of √39 ≈ 6.245.
How many triangles fit a = 7, b = 10 and A = 40°? Two: angle B can be 66.67° or 113.33°.
Angles, area and type of 22 triangles solved from their three sides
| Sides a, b, c | Angle A | Angle B | Angle C | Area | Type |
|---|---|---|---|---|---|
| 3, 4, 5 | 36.87° | 53.13° | 90.00° | 6.00 | scalene, right |
| 5, 12, 13 | 22.62° | 67.38° | 90.00° | 30.00 | scalene, right |
| 8, 15, 17 | 28.07° | 61.93° | 90.00° | 60.00 | scalene, right |
| 7, 24, 25 | 16.26° | 73.74° | 90.00° | 84.00 | scalene, right |
| 1, 1, 1 | 60.00° | 60.00° | 60.00° | 0.43 | equilateral, acute |
| 2, 2, 3 | 41.41° | 41.41° | 97.18° | 1.98 | isosceles, obtuse |
| 5, 5, 8 | 36.87° | 36.87° | 106.26° | 12.00 | isosceles, obtuse |
| 6, 6, 6 | 60.00° | 60.00° | 60.00° | 15.59 | equilateral, acute |
| 4, 5, 6 | 41.41° | 55.77° | 82.82° | 9.92 | scalene, acute |
| 5, 6, 7 | 44.42° | 57.12° | 78.46° | 14.70 | scalene, acute |
| 7, 8, 9 | 48.19° | 58.41° | 73.40° | 26.83 | scalene, acute |
| 2, 3, 4 | 28.96° | 46.57° | 104.48° | 2.90 | scalene, obtuse |
| 3, 5, 7 | 21.79° | 38.21° | 120.00° | 6.50 | scalene, obtuse |
| 6, 7, 12 | 20.85° | 24.53° | 134.62° | 14.95 | scalene, obtuse |
| 9, 10, 17 | 25.06° | 28.07° | 126.87° | 36.00 | scalene, obtuse |
| 10, 10, 10 | 60.00° | 60.00° | 60.00° | 43.30 | equilateral, acute |
| 13, 14, 15 | 53.13° | 59.49° | 67.38° | 84.00 | scalene, acute |
| 1, 1, 1.4142 | 45.00° | 45.00° | 90.00° | 0.50 | isosceles, right |
| 1, 1.7321, 2 | 30.00° | 60.00° | 90.00° | 0.87 | scalene, right |
| 4, 7, 10 | 18.19° | 33.12° | 128.68° | 10.93 | scalene, obtuse |
| 10, 13, 20 | 25.59° | 34.16° | 120.25° | 56.15 | scalene, obtuse |
| 11, 12, 13 | 52.02° | 59.30° | 68.68° | 61.48 | scalene, acute |
Tips
- Name the sides by their opposite angle. Most mistakes come from putting an angle next to the wrong side.
- Check the triangle inequality. Each side must be shorter than the other two added together.
- For right triangles, use the right triangle calculator. It takes two values instead of three.
FAQ
What do SSS, SAS, ASA and AAS mean?
They name which three parts you know, read round the triangle. SSS is three sides. SAS is two sides and the angle between them. ASA is two angles and the side between them. AAS is two angles and a side that is not between them. Each one fixes exactly one triangle.
Why can SSA give two answers?
With two sides and an angle not between them, the second side can swing to meet the base at two points. The law of sines gives sin B, and both B and 180° − B share that sine. If both leave room for the third angle, two different triangles fit the same three facts.
How do you find the angles of a triangle from its sides?
Use the law of cosines rearranged: cos A = (b² + c² − a²) ÷ 2bc. Take the inverse cosine to get A, repeat for B, then C = 180° − A − B. Choose Three sides (SSS) above and the calculator does every step and shows the numbers it used.
How do you tell if a triangle is acute, right or obtuse from its sides?
Square the longest side and compare it with the sum of the squares of the other two. Equal means a right triangle, smaller means acute and larger means obtuse. Sides 7, 8 and 9 give 81 against 113, so that triangle is acute.
See also: all Algebra and Geometry tools, Arithmetic, Number Theory, Statistics, Trigonometry and Calculus.
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