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Triangle Calculator
Enter the three values you know, whether three sides, two sides and the angle between them, two angles and a side, or two values of a right triangle, and the calculator solves the rest: every side and angle, the area, perimeter, heights, medians and the triangle type, with a to-scale drawing.
What do you know?
Side a is opposite angle A, side b opposite angle B, side c opposite angle C.
°
°
°
Enter your values to see the result.
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How to use the Triangle Calculator
- Choose what you know: SSS, SAS, ASA, AAS or right triangle.
- Enter the values. Side a is opposite angle A, b opposite B and c opposite C. Angles are in degrees.
- Read the solution: area, perimeter, all sides and angles and the type appear instantly.
- Check the drawing and steps to see how each value was found.
- Download the table of heights and medians or copy the results.
Triangle formulas used
| Rule | Formula | Used for |
|---|---|---|
| Angle sum | A + B + C = 180° | Third angle (ASA, AAS, right) |
| Law of cosines | c² = a² + b² − 2ab·cos C | SSS angles, SAS third side |
| Law of sines | a / sin A = b / sin B = c / sin C | ASA and AAS sides |
| Pythagorean theorem | a² + b² = c² (C = 90°) | Right triangles |
| Area | ½ab·sin C = √(s(s−a)(s−b)(s−c)) | Every case |
| Height to side a | hₐ = 2 · Area ÷ a | Altitudes |
| Median to side a | mₐ = ½√(2b² + 2c² − a²) | Medians |
| Inradius / circumradius | r = Area ÷ s, R = a ÷ (2 sin A) | Inscribed and circumscribed circles |
Special right triangles
| Triangle | Angles | Side ratio | Area (shortest leg = 1) |
|---|---|---|---|
| 45-45-90 | 45°, 45°, 90° | 1 : 1 : √2 (≈ 1.4142) | 0.5 |
| 30-60-90 | 30°, 60°, 90° | 1 : √3 : 2 (≈ 1.7321) | ≈ 0.8660 |
| 3-4-5 | ≈ 36.87°, 53.13°, 90° | 3 : 4 : 5 | 6 (for legs 3 and 4) |
| 5-12-13 | ≈ 22.62°, 67.38°, 90° | 5 : 12 : 13 | 30 (for legs 5 and 12) |
Naming convention: vertices are capital letters and each side uses the lower-case letter of the vertex opposite it, so side a never touches vertex A. In right-triangle mode the right angle is C, so c is the hypotenuse.
Frequently asked questions
What do SSS, SAS, ASA and AAS mean?
They describe which three values you know. SSS: three sides. SAS: two sides and the angle between them. ASA: two angles and the side between them. AAS: two angles and a side that is not between them. Each of these fixes exactly one triangle.
Why is SSA (two sides and a non-included angle) not here?
Two sides and an angle that is not between them can match zero, one or two different triangles (the ambiguous case), so it does not determine a unique answer. Use SAS if you know the angle between the two sides.
What is the law of cosines?
c² = a² + b² − 2ab·cos C. It generalizes the Pythagorean theorem to any triangle and is used to find a side from two sides and the included angle (SAS) or an angle from three sides (SSS).
What is the law of sines?
a ÷ sin A = b ÷ sin B = c ÷ sin C = 2R, where R is the circumradius. Once you know two angles and one side (ASA or AAS), it gives the other two sides.
How is the area calculated?
The calculator uses Area = ½ · a · b · sin C, which works for any triangle once two sides and the angle between them are known. For three sides it gives the same result as Heron's formula.
Why do my three sides not make a triangle?
Every side must be shorter than the other two added together (the triangle inequality). Sides 2, 3 and 6 fail because 2 + 3 is less than 6, so they cannot close into a triangle.
How are triangles classified?
By sides: equilateral (all equal), isosceles (two equal) or scalene (none equal). By angles: acute (all under 90°), right (one exactly 90°) or obtuse (one over 90°). The angles of any triangle always add up to 180°.
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