Enter side lengths
Provide three sides (a, b, c). The solver validates triangle inequality and computes angles and measures.
Example data
Try these side sets to see different triangle types and outputs.
| Case | a | b | c | Expected type |
|---|---|---|---|---|
| 1 | 3 | 4 | 5 | Right, scalene |
| 2 | 5 | 5 | 5 | Equilateral |
| 3 | 6 | 6 | 10 | Obtuse, isosceles |
| 4 | 7 | 8 | 9 | Acute, scalene |
Formulas used
- Triangle inequality: a+b>c, a+c>b, b+c>a.
- Law of Cosines: cos A = (b² + c² − a²) / (2bc), similarly for B and C.
- Heron’s formula: p = (a+b+c)/2, Area K = √(p(p−a)(p−b)(p−c)).
- Heights: ha=2K/a, hb=2K/b, hc=2K/c.
- Inradius: r = K/p. Circumradius: R = abc/(4K).
- Medians: ma=0.5√(2b²+2c²−a²), similarly for b and c.
- Angle bisectors: ℓa=√(bc((b+c)²−a²))/(b+c), similarly for b and c.
How to use this calculator
- Enter side lengths a, b, and c using the same unit.
- Optionally set units and the number of decimals to display.
- Click Solve triangle to compute angles and measures.
- If the triangle is invalid, adjust sides to satisfy a+b>c, etc.
- Use Download CSV or Download PDF to save the results.
Why SSS validation matters in practical work
SSS solving begins with the triangle inequality because real inputs often come from tapes, sensors, or CAD exports. If any pair sum is not greater than the third side, the geometry collapses and angles are undefined. This calculator flags the issue immediately, helping you spot typos, unit mixing, or tolerance drift before continuing.
Angle computation with reliable safeguards
Angles are computed with the Law of Cosines, converting side lengths into cosine values. Near-degenerate data can push a cosine slightly outside the valid range due to rounding, so the solver clamps values between −1 and 1 before applying arccos. An angle-sum check is shown so you can confirm the result is close to 180°.
Area, heights, and radii from a single core value
Heron’s formula uses the semiperimeter p to produce area K from three sides only. Once K is known, heights follow directly from h = 2K/side, which supports clearance checks and layout work. The inradius r = K/p helps inscribed-circle designs, and the circumradius R = abc/(4K) supports chord and arc constructions.
Classification as a quick quality signal
The solver classifies the triangle by side equality and by angle type. Angle type is determined by comparing the largest squared side to the sum of the other squared sides, highlighting right, acute, or obtuse geometry. In practice, an unexpected obtuse result can reveal skewed measurements. Pair this with the plot to visually validate shape and scale.
Precision, rounding, and export-ready reporting
Different workflows require different rounding. Field notes may use two decimals, while computational geometry may need four or more. The decimals control standardizes outputs and reduces copy errors. CSV export fits spreadsheets and audits, while PDF export provides a compact report for clients, students, or documentation. Keep the unit label consistent to compare batches correctly.
Typical datasets and benchmark examples
Common benchmarks include (3, 4, 5) to verify right triangles and equal-side sets to test symmetry. Near-degenerate sets are useful for checking stability when one side is almost the sum of the others. In CAD, sides may come from coordinate distances, so small floating differences are normal; clamping and checks keep results dependable across repeated runs, without changing your underlying input values.
FAQs
1) What does SSS mean for triangles?
SSS means you know all three side lengths. With those values, you can uniquely determine a valid triangle’s angles, area, and many derived measures.
2) Why does the calculator reject some inputs?
If any two sides add up to less than or equal to the third, the triangle cannot exist. The solver checks this triangle inequality before computing angles.
3) How are angles computed from sides?
Angles come from the Law of Cosines. It converts side lengths into cosine values, then uses arccos to obtain angles in degrees.
4) What is the difference between inradius and circumradius?
The inradius is the radius of the inscribed circle touching all sides. The circumradius is the radius of the circumscribed circle passing through all vertices.
5) Does the plotted triangle preserve scale?
Yes. The plot constructs coordinates using your sides, placing one side on the x-axis and solving the third vertex position. The aspect ratio is locked for accuracy.
6) What units should I use?
Any consistent unit works, such as cm or meters. The calculator does not convert units; it simply carries your chosen label into outputs and exports.