This triangle solver computes the area, perimeter, and all three interior angles of any valid triangle given the lengths of its three sides ($a, b, c$). It automatically checks the fundamental Triangle Inequality Theorem before applying Heron's formula and the Law of Cosines to solve the complete geometric profile.

### Fundamental Geometry & Formulas

1. **Triangle Inequality Verification:** For any valid triangle, the sum of any two sides must strictly exceed the third side: $$a + b > c, \quad a + c > b, \quad b + c > a$$ If any condition fails, the sides cannot connect in Euclidean space. 2. **Semi-Perimeter & Heron's Formula:** $$s = \frac{a + b + c}{2}$$ $$\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$$ 3. **Interior Angles via the Law of Cosines:** $$\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos(B) = \frac{a^2 + c^2 - b^2}{2ac}, \quad C = 180^\circ - A - B$$

### Worked Geometric Example

Given sides $a = 5$, $b = 6$, and $c = 7$: - **Perimeter:** $5 + 6 + 7 = 18$ units - **Semi-Perimeter ($s$):** $18 / 2 = 9$ units - **Heron's Area:** $$\text{Area} = \sqrt{9 \times (9 - 5) \times (9 - 6) \times (9 - 7)} = \sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} \approx 14.70\text{ sq units}$$ - **Angles:** - $\text{Angle } A = \arccos\left(\frac{36 + 49 - 25}{2 \times 6 \times 7}\right) = \arccos\left(\frac{60}{84}\right) \approx 44.4^\circ$ - $\text{Angle } B = \arccos\left(\frac{25 + 49 - 36}{2 \times 5 \times 7}\right) = \arccos\left(\frac{38}{70}\right) \approx 57.1^\circ$ - $\text{Angle } C = 180^\circ - 44.4^\circ - 57.1^\circ = 78.5^\circ$ - Sum of angles: $44.4^\circ + 57.1^\circ + 78.5^\circ = 180.0^\circ$

### Pure Client-Side Computation

All calculations execute instantly within your browser using native JavaScript floating-point trigonometry. No geometry inputs are logged or sent to remote servers.