Triangle Calculator from Three Sides (SSS)
Enter all three side lengths to find every angle via the law of cosines, and the area, perimeter, inradius, and circumradius via Heron's formula.
Computed with Heron's formula from all three sides
- 1
Semi-perimeter
s = (5 + 7 + 8) ÷ 2 = 10 - 2
Product under the radical
s(s−a)(s−b)(s−c) = 10 × 5 × 3 × 2 = 300All four factors must be positive — this is guaranteed by the triangle inequality. - 3
Area (Heron's formula)
√300 = 17.3205
How does this calculator work?
Given sides a, b, c satisfying the triangle inequality, find cos A = (b²+c²−a²)/(2bc) and similarly for B and C. Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 (Heron's formula). Inradius r = area/s; circumradius R = abc/(4·area). All three angles plus area and perimeter are computed instantly.
Formula
How this is calculated
When all three sides of a triangle are known (the SSS case), the law of cosines gives each angle directly. For angle A opposite side a: cos A = (b² + c² − a²) / (2bc). Rearranging for B and C gives the complete angle set. The sides must first satisfy the triangle inequality — the sum of any two sides must strictly exceed the third — otherwise no real triangle exists.
The area is found with Heron's formula: compute the semi-perimeter s = (a + b + c) / 2, then area = √(s(s−a)(s−b)(s−c)). This formula requires no angles and is the standard approach for the SSS case. The inradius (r = area / s) is the radius of the largest inscribed circle, and the circumradius (R = abc / (4 × area)) is the radius of the circumscribed circle that passes through all three vertices.
All results share the same length unit as the input sides. Degenerate inputs — a side of zero, or two sides that sum exactly to the third — produce no result rather than a flat (zero-area) triangle.
Frequently asked questions
If the sum of the two shorter sides is less than or equal to the longest side, no real triangle exists with those dimensions. The calculator returns no result. Adjust your values so that a + b > c, a + c > b, and b + c > a.
Heron's formula (area = √(s(s−a)(s−b)(s−c))) computes the area of any triangle from its three sides alone, without needing the height. It is numerically stable for most triangles and is the standard for the SSS case, avoiding the need to first compute angles.
The inradius r is the radius of the inscribed circle — the largest circle fitting inside the triangle — equal to area / s. The circumradius R is the radius of the circumscribed circle through all three vertices, equal to abc / (4 × area). Both radii appear in many geometry proofs and construction problems.
Also known as
TG we-Calculate Editorial Team. (2026). Triangle Calculator from Three Sides (SSS) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/abc-triangle-calculator
TG we-Calculate Editorial Team. "Triangle Calculator from Three Sides (SSS)." TG we-Calculate. 2026. https://we-calculate.com/calculator/abc-triangle-calculator.
TG we-Calculate Editorial Team, "Triangle Calculator from Three Sides (SSS)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/abc-triangle-calculator
@misc{wecalculate_abc_triangle_calculator, title = {Triangle Calculator from Three Sides (SSS)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/abc-triangle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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