Triangle Calculator Calculate sides, angles, area, perimeter, and other properties of triangles — with visual representat...

Triangle Calculator

Calculate sides, angles, area, perimeter, and other properties of triangles — with visual representation and step-by-step solutions.

SSS (3 Sides)

SAS (2 Sides, Included Angle)

ASA (2 Angles, Included Side)

Right Triangle

Side a

Side b

Side c

Side a

Angle C (°)

Side b

Angle A (°)

Side c

Angle B (°)

Leg a

Leg b

Hypotenuse c

3.00

Side a

4.00

Side b

5.00

Side c

36.87°

Angle A

53.13°

Angle B

90.00°

Angle C

6.00

Area

12.00

Perimeter

Triangle Formulas

Law of Cosines

c² = a² + b² - 2ab·cos(C)

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C)

Area (Heron's)

A = √[s(s-a)(s-b)(s-c)]

Pythagorean

a² + b² = c²

Triangle Classification

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Triangle Visualization

Triangle Properties Explained

A triangle is a polygon with three sides and three angles. The sum of interior angles is always 180°.

Key Properties**:

• Sides**: Three line segments that form the triangle (a, b, c)
• Angles**: Three interior angles opposite to the sides (A, B, C)
• Area**: Space enclosed by the triangle
• Perimeter**: Sum of all three sides

Triangle Types**:

• By Sides**: Equilateral (all equal), Isosceles (two equal), Scalene (all different)
• By Angles**: Acute (all < 90°), Right (one = 90°), Obtuse (one > 90°)

✅ Fundamental Theorem**: The sum of any two sides must be greater than the third side (Triangle Inequality).

Common Triangle Calculation Mistakes

⚠️ Avoid these frequent errors:

• Triangle inequality violation**: Sides that don't satisfy a + b > c
• Angle sum error**: Angles not adding up to 180°
• Ambiguous case (SSA)**: Two possible triangles for certain inputs
• Unit inconsistency**: Mixing degrees and radians
• Right triangle assumption**: Assuming a triangle is right-angled without verification

✅ Best Practices**:

• Always verify triangle inequality before calculations
• Check that angles sum to 180° in your results
• For SSA cases, consider both possible solutions
• Use appropriate formulas based on given information
• Draw a diagram to visualize the problem

Real-World Applications

Triangle calculations are essential in:

• Construction**: Roof pitches, stair angles, structural analysis
• Navigation**: GPS triangulation, surveying, map reading
• Engineering**: Force analysis, mechanical linkages, bridge design
• Computer Graphics**: 3D modeling, rendering, game development
• Astronomy**: Parallax measurements, celestial navigation

📊 Example Use Cases**:

• Roof design**: 3-4-5 triangle ensures perfect right angle for carpentry
• Navigation**: Two bearings and distance between points determine position
• Surveying**: Measure two angles and included side to find area of land plot
• Physics**: Resolve forces into components using right triangles

How to Use This Calculator

➡️ SSS (3 Sides)

"Sides 3, 4, 5" → Angles: 36.87°, 53.13°, 90°, Area: 6.00

➡️ SAS (2 Sides, Included Angle)

"Sides 5, 7 with 60° angle" → Third side: 6.24, Area: 15.16

➡️ ASA (2 Angles, Included Side)

"Angles 45°, 60° with side 10" → Third angle: 75°, Other sides: 7.32, 8.97

➡️ Right Triangle

"Legs 3, 4" → Hypotenuse: 5.00, Area: 6.00, Angles: 36.87°, 53.13°

Note: All angle calculations use degrees. Results are rounded to 2 decimal places. The calculator validates triangle inequality and angle sum constraints.