SSS Triangle Calculator (Side-Side-Side)
Enter all three side lengths. The calculator applies the law of cosines to find every angle and Heron's formula to find area, then derives perimeter, inradius, circumradius, and all three altitudes.
Triangle type: Acute
- 1
Semi-perimeter s
(5 + 7 + 8) ÷ 2 = 10 - 2
Heron's product s(s−a)(s−b)(s−c)
10 × 5 × 3 × 2 = 300 - 3
Area = √(product)
√300 = 17.3205
How does this calculator work?
Given all three sides a, b, c: check the triangle inequality (each < sum of other two), then find A = arccos((b²+c²−a²)/2bc), B similarly, C = 180°−A−B. Area = √(s(s−a)(s−b)(s−c)) with s = (a+b+c)/2, circumradius R = abc/4A, inradius r = A/s. The SSS case is always unique — no ambiguity.
Formula
How this is calculated
When all three sides are known the triangle is uniquely determined — the SSS case has no ambiguity. The first step is to verify the triangle inequality: each side must be strictly less than the sum of the other two. If this fails, no triangle exists.
Angles are recovered by the law of cosines. Rearranging the standard form c² = a² + b² − 2ab cos(C) gives cos(A) = (b² + c² − a²) / (2bc), and similarly for B. The third angle C = 180° − A − B avoids accumulating the rounding error that a separate arccos would introduce for the third angle. All three arccos calls return values in [0°, 180°], so obtuse triangles are handled correctly without any sign-check.
Area comes from Heron's formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 is the semi-perimeter. The circumradius (radius of the circumscribed circle) is R = abc / (4·Area), and the inradius (inscribed circle) is r = Area / s. The altitude to side a is 2·Area / a, and similarly for b and c.
Frequently asked questions
Yes — three positive side lengths that satisfy the triangle inequality determine a triangle up to congruence. There is no ambiguous case in SSS (unlike SSA). The law of cosines always returns a unique angle in [0°, 180°].
For any valid triangle, each side must be strictly shorter than the sum of the other two sides (a < b + c, b < a + c, c < a + b). If this fails for any combination, the sides cannot form a closed triangle.
Heron's formula computes the area of a triangle purely from its three side lengths: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter. It is named after the ancient Greek mathematician Heron of Alexandria.
Also known as
TG we-Calculate Editorial Team. (2026). SSS Triangle Calculator (Side-Side-Side) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/sss-triangle-calculator
TG we-Calculate Editorial Team. "SSS Triangle Calculator (Side-Side-Side)." TG we-Calculate. 2026. https://we-calculate.com/calculator/sss-triangle-calculator.
TG we-Calculate Editorial Team, "SSS Triangle Calculator (Side-Side-Side)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/sss-triangle-calculator
@misc{wecalculate_sss_triangle_calculator, title = {SSS Triangle Calculator (Side-Side-Side)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/sss-triangle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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