Triangle Calculator
Enter any three sides or angles of a triangle, at least one a side. Solves SSS, SAS, ASA, AAS and both SSA triangles, with area, perimeter and heights. How it works
Use the Triangle Calculator
About the Triangle Calculator
A 3-4-5 triangle has angles of 36.8699°, 53.1301° and 90° and an area of 6. This triangle calculator solves any triangle from three known values, at least one of them a side: it finds the missing sides and angles, then the area, perimeter, three heights, inradius and circumradius.
The five cases and how each is solved
| Case | What you know | How it is solved | Example input | Solved triangle |
|---|---|---|---|---|
| SSS | Three sides | Law of cosines for each angle | a = 7, b = 8, c = 9 | A = 48.1897°, B = 58.4119°, C = 73.3985°, area 26.8328 |
| SAS | Two sides and the angle between them | Law of cosines for the third side | a = 5, b = 7, C = 60° | c = 6.245, A = 43.8979°, B = 76.1021°, area 15.1554 |
| ASA | Two angles and the side between them | Angle sum, then law of sines | A = 50°, B = 60°, c = 10 | a = 8.15207, b = 9.21605, C = 70°, area 35.2995 |
| AAS | Two angles and a side not between them | Angle sum, then law of sines | A = 40°, B = 75°, a = 8 | b = 12.0217, c = 11.2797, C = 65°, area 43.5815 |
| SSA | Two sides and an angle opposite one of them | Law of sines; 0, 1 or 2 triangles | a = 6, b = 10, A = 30° | c = 11.9769, B = 56.4427°, C = 93.5573°, area 29.9422; or c = 5.34363, B = 123.557°, C = 26.4427°, area 13.3591 |
Angle A is opposite side a, B is opposite b and C is opposite c. Any lettering works: sides b and c with angle A is SAS just as a, b and C is.
SSA, the ambiguous case: triangles that fit A = 30° and b = 10
| Side a | Triangles | Result |
|---|---|---|
| 4 | 0 | None: a would have to be at least b × sin A = 5 |
| 5 | 1 | B = 90°, C = 60°, c = 8.66025 |
| 6 | 2 | B = 56.4427°, C = 93.5573°, c = 11.9769; or B = 123.557°, C = 26.4427°, c = 5.34363 |
| 10 | 1 | B = 30°, C = 120°, c = 17.3205 |
| 12 | 1 | B = 24.6243°, C = 125.376°, c = 19.569 |
Two triangles appear only when a is longer than b × sin A and shorter than b.
Special triangles: sides, angles and area
| Triangle | Sides | Angles | Area | Height on the longest side |
|---|---|---|---|---|
| 3-4-5 | 3, 4, 5 | 36.8699°, 53.1301°, 90° | 6 | 2.4 |
| 5-12-13 | 5, 12, 13 | 22.6199°, 67.3801°, 90° | 30 | 4.61538 |
| 8-15-17 | 8, 15, 17 | 28.0725°, 61.9275°, 90° | 60 | 7.05882 |
| 7-24-25 | 7, 24, 25 | 16.2602°, 73.7398°, 90° | 84 | 6.72 |
| 45-45-90, legs of 1 | 1, 1, 1.41421 | 45°, 45°, 90° | 0.5 | 0.707107 |
| 30-60-90, hypotenuse of 2 | 1, 1.73205, 2 | 30°, 60°, 90° | 0.866025 | 0.866025 |
| Equilateral, sides of 1 | 1, 1, 1 | 60°, 60°, 60° | 0.433013 | 0.866025 |
Multiply every side by the same factor and the angles stay as they are, while the area grows by the square of the factor.
How to use this tool
- Choose degrees or radians for the angles.
- Type the three values you know and leave the other fields empty. Angle A is opposite side a, B is opposite b and C is opposite c.
- Read the case the tool detected (SSS, SAS, ASA, AAS or SSA) and the law it used, then the solved triangle. An SSA input with two solutions shows both.
Frequently asked questions
- How do I find the angles of a triangle from its three sides?
- Use the law of cosines: cos C = (a² + b² − c²) ÷ (2ab), and the same pattern for the other two angles. Sides 7, 8 and 9 give angles of 48.1897°, 58.4119° and 73.3985°. Type the three sides and leave the angle fields empty.
- How do I solve a triangle with two sides and the angle between them (SAS)?
- The law of cosines gives the third side: c² = a² + b² − 2ab × cos C. With a = 5, b = 7 and C = 60°, c is 6.245 and the other two angles are 43.8979° and 76.1021°.
- What is the ambiguous case (SSA)?
- Two sides and an angle that is not between them can fit zero, one or two triangles. With A = 30° and b = 10, a = 4 is too short to reach and gives no triangle, a = 5 gives one right triangle, a = 6 gives two (c = 11.9769 or c = 5.34363) and a = 12 gives one.
- How do I find the area of a triangle from three sides?
- Heron’s formula: with s = (a + b + c) ÷ 2, the area is √(s(s − a)(s − b)(s − c)). For sides 5, 6 and 7, s is 9 and the area is √216 = 14.6969.
- Can I solve a triangle from three angles?
- No. Three angles fix the shape but not the size, so the tool asks for at least one side. Angles of 30° and 60° with a hypotenuse of 2 give legs of 1 and 1.73205; with a hypotenuse of 4 the legs are 2 and 3.4641.
- Why does it say my sides cannot form a triangle?
- The two shorter sides have to add up to more than the longest. Sides 1, 2 and 4 fail because 1 + 2 is less than 4. Sides 1, 2 and 3 make only a flat line with area 0, which the tool reports as a degenerate triangle.
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