Triangle Calculator
Solve a plane triangle from SSS, SAS, or two angles and an opposite side, with unit conversion.
Content updated August 3, 2026
This solves a plane Euclidean triangle. All entered lengths must use the selected input unit, and every angle input and output is in degrees. Measurements near a degenerate triangle can be especially sensitive to input precision.
Solve a plane triangle from three sides (SSS), two sides and their included angle (SAS), or two angles with the side opposite angle A (ASA/AAS). The result includes every side and angle, area, perimeter, semiperimeter, inradius, circumradius, and side and angle classifications.
Enter every length in one input unit and choose a separate output unit. Lengths convert linearly while area converts by the square of that factor; angles remain in degrees.
The Triangle formula
SSS: K = √[s(s−a)(s−b)(s−c)]; SAS: c² = a²+b²−2ab cos C; ASA/AAS: C = 180°−A−B and b/sin B = c/sin C = a/sin Aa, b, c are sides · A, B, C are their opposite angles · s = (a+b+c)/2 · K = area. Inradius = K/s and circumradius = abc/(4K).
Worked example
For sides 3, 4, and 5, the semiperimeter is 6 and Heron's formula gives area √(6×3×2×1) = 6. The angles are approximately 36.87°, 53.13°, and 90°, so it is a scalene right triangle.
Assumptions, rounding, and limitations
Assumptions
- The figure is a nondegenerate triangle in a Euclidean plane.
- All entered lengths share the selected input unit and every angle is in degrees.
- SAS uses angle C between sides a and b; ASA/AAS uses side a opposite angle A.
- Input measurements are treated as exact values at the precision entered.
Rounding: The engine retains floating-point precision through solving and unit conversion. Length and area displays show up to eight decimal places and angle displays up to six decimal places.
Limitations
- Each entered or calculated length must be positive and no more than 1 trillion; angles must be positive and form a total below 180° where entered.
- Does not solve ambiguous SSA input, spherical or hyperbolic triangles, coordinates, bearings, uncertainty propagation, or measurement tolerances.
- Nearly degenerate triangles are sensitive to input precision, and floating-point results can differ slightly from exact symbolic values.
Sources
- SI Brochure, Section 5: Units outside the SI — National Institute of Standards and Technology
- Trigonometric Functions — NIST Digital Library of Mathematical Functions
Choosing SSS, SAS, or ASA/AAS
Use SSS when all three side lengths are known. Use SAS when sides a and b and their included angle C are known. Use ASA/AAS when angles A and B and side a opposite angle A are known; the third angle is 180° − A − B.
The input combinations are unambiguous. The calculator does not offer side-side-angle because some SSA measurements can describe zero, one, or two different triangles.
How the triangle is solved
SSS uses a numerically stable form of Heron's formula for area. SAS first finds side c with the law of cosines, expressed in a stable half-angle form. ASA/AAS uses the angle sum and law of sines to find the remaining sides.
After the sides are known, angles are recovered from side relationships, perimeter is the side sum, inradius is area divided by semiperimeter, and circumradius is abc divided by four times the area.
Frequently asked questions
▶What does SSS mean?
SSS means all three side lengths are known. The lengths must be positive and each pair must sum to more than the remaining side.
▶Which angle do I enter for SAS?
Enter angle C, the included angle between side a and side b. It must be greater than 0° and less than 180°.
▶Which side is used in ASA/AAS mode?
Enter side a, which is opposite angle A. Together with angles A and B, this uniquely sets the triangle's scale.
▶Why is a degenerate triangle rejected?
If two side lengths sum to the third, the shape collapses to a line and has zero area, so it is not a nondegenerate plane triangle.
▶How are area units converted?
Area is multiplied by the square of the length-conversion factor. For example, 1 square metre equals 10,000 square centimetres.
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Open calculator →Disclaimer: Triangle Calculator results are estimates for general information and education only, and are not financial, tax, legal or medical advice. Verify important decisions with a qualified professional.