Triangle Calculator
Free triangle calculator. Find all sides, angles, area and perimeter of any triangle using Law of Cosines and Law of Sines.
Triangle Calculator
Enter any 3 known values (at least one must be a side). Leave unknowns empty.
Comprehensive Guide: Triangle Solving, Law of Sines & Law of Cosines
Solve any right or oblique triangle using the Pythagorean theorem, SOH-CAH-TOA, Law of Sines, Law of Cosines, and Heron’s formula.
1. The Fundamental Geometry of Triangles & Classification
A triangle is the simplest polygon in Euclidean 2D space, formed by three non-collinear vertices connected by straight line segments. The universal axiom of Euclidean triangle geometry states that the three interior angles always sum to exactly 180 degrees: α + β + γ = 180° (or π radians).
2. Right Triangles: SOH CAH TOA & Pythagorean Theorem
For triangles possessing one 90° right angle, trigonometry is defined by side ratios relative to an acute angle θ:
sin(θ) = Opp / Hyp | cos(θ) = Adj / Hyp | tan(θ) = Opp / Adj
3. Oblique Triangles: The Law of Sines & Law of Cosines
For general triangles lacking right angles, two universal trigonometric laws apply:
The Law of Sines (Used for ASA, AAS, and SSA)
The Law of Cosines (Used for SAS and SSS)
4. Graduated Step-by-Step Worked Problems
Problem 1: SSS Triangle (Sides a = 7, b = 8, c = 9)
Step 1: Compute angle γ using Law of Cosines:
cos(γ) = (7² + 8² − 9²) / (2 × 7 × 8) = (49 + 64 − 81) / 112 = 32 / 112 ≈ 0.2857.
γ = arccos(0.2857) ≈ 73.40°.
Step 2: Compute angle α using Law of Sines:
sin(α) = (a · sin(γ)) / c = (7 · sin(73.40°)) / 9 ≈ (7 × 0.9583) / 9 ≈ 0.7454.
α = arcsin(0.7454) ≈ 48.19°.
Step 3: Compute remaining angle β:
β = 180° − 73.40° − 48.19° = 58.41°.
Verification: 48.19° + 58.41° + 73.40° = 180.00° ✓
Problem 2: Area Calculation via SAS Trigonometric Formula
Find the area of a triangle with sides a = 12 cm, b = 15 cm and included angle γ = 42°:
• Area formula: A = (1/2) · a · b · sin(γ).
• A = 0.5 × 12 × 15 × sin(42°) = 90 × 0.66913 ≈ 60.22 cm².
Result: Area is 60.22 cm².
5. The Ambiguous Case (SSA): Zero, One, or Two Triangles
When two sides and an opposite acute angle are given (SSA), the configuration may yield:
- No triangle: If
a < h = b·sin(α)(sideais too short to reach base). - One right triangle: If
a = h. - Two distinct triangles: If
h < a < b(one acute triangle, one obtuse triangle). - One unique triangle: If
a ≥ b.
6. Frequently Asked Questions (FAQ)
What is the Triangle Inequality Theorem?
The sum of lengths of any two sides of a valid triangle must be strictly greater than the length of the third side: a + b > c, a + c > b, and b + c > a. If a + b ≤ c, the segments cannot meet to form an enclosed triangle.
When do I use the Law of Sines vs. Law of Cosines?
Use the Law of Sines when you have an angle and its opposite side (AAS, ASA, or SSA). Use the Law of Cosines when you have all three sides (SSS) or two sides and the included angle between them (SAS).
Why are triangular trusses used so heavily in bridges and roofs?
Triangles are the only geometrically rigid 2D polygon. Once the three side lengths are fixed, the interior angles cannot deform or flex without physically bending or breaking the constituent beams.