Triangle Calculator Online
Calculate triangle area via Heron's formula, perimeter, all 3 angles, and inradius/circumradius.
Input Parameters
Live Result
Details
Steps
Side lengths: a = 3, b = 4, c = 5
Perimeter: P = a + b + c = 3 + 4 + 5 = 12
Semi-perimeter: s = P / 2 = 6
Heron's Formula: Area = √(s(s - a)(s - b)(s - c))
Factor evaluation: √(6 × 3 × 2 × 1) = √(36) = 6
Angles (Law of Cosines): ∠A ≈ 36.8699°, ∠B ≈ 53.1301°, ∠C ≈ 90° (Sum = 180°)
Inradius (r = S/s): 1, Circumradius (R = abc/4S): 2.5
Quick Answer & Understanding Your Result
A Triangle Calculator solves complete geometric properties of any triangle given its 3 side lengths (a, b, c): Area is calculated using Heron's Formula: Area = √(s(s - a)(s - b)(s - c)), where s = (a + b + c) / 2 is the semi-perimeter. Angles are solved via the Law of Cosines.
Triangle Classifications
- By Sides: Equilateral (3 equal sides), Isosceles (2 equal sides), or Scalene (all sides unequal).
- By Angles: Right Triangle (a² + b² = c², one 90° angle), Acute (< 90°), or Obtuse (> 90°).
- Inradius (r): Radius of inscribed circle: r = Area / s.
- Circumradius (R): Radius of circumscribed circle: R = abc / (4 × Area).
The Triangle Inequality Theorem
- Mandatory Validity: A real triangle can only exist if the sum of any two sides is strictly greater than the third side: a + b > c, a + c > b, and b + c > a.
Enter your side lengths above to compute area, perimeter, all three angles, and complete classifications instantly.
Formula & Calculation Logic
Mathematical FormulaHeron's formula computes area directly from 3 side lengths without requiring height. The Law of Cosines determines internal angles.
Step-by-Step Guide
Enter 3 Sides
Input positive lengths for sides a, b, and c.
Validate Inequality
Check that sum of any two sides is strictly greater than the third.
Calculate Area & Perimeter
Compute semi-perimeter s and apply Heron's formula for area.
Solve Angles & Radii
Determine all three angles via Law of Cosines, plus inradius and circumradius.
Worked Examples Table
| Example | Input Values | Calculated Result | Explanation |
|---|---|---|---|
| Right Triangle 3-4-5 | s = 6. Area = √(6 × 3 × 2 × 1) = √36 = 6. Angle C = 90°. | ||
| Equilateral Triangle 6-6-6 | All 3 sides equal 6. Angles are 60° each. | ||
| Isosceles Triangle 5-5-6 | s = 8. Area = √(8 × 3 × 3 × 2) = √144 = 12. |
Frequently Asked Questions
Evidence-Based Methodology & Authoritative References
Wolfram MathWorld - Triangle
NIST Digital Library of Mathematical Functions - Triangles
Standard: ISO / W3C / IETF Authoritative Domain Standard
Formula: s = (a + b + c) / 2
Last Reviewed: August 21, 2026
Primary Source: Wolfram MathWorld - Triangle
Reference Document: NIST Digital Library of Mathematical Functions - Triangles
Computation Privacy: 100% client-side 0ms computation, zero data retention
Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.
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Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.