Geometry & Mensuration

Free Triangle Calculator — Area, Angles & Pythagoras Solver

Use our free triangle calculator to solve right and oblique triangles using Pythagorean theorem, Heron's formula, Law of Sines, and Law of Cosines with step-by-step math.

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Triangle Dimensions, Angles & Area Solver

Select your calculation method to solve any triangle

Triangle Area
6.0000 cm²
Perimeter
12.0000 cm
Classification
Right Scalene
Angle A
36.87°
Angle B
53.13°
Angle C
90.00°
a=3 b=4 c=5
• Sides: a = 3 cm, b = 4 cm, c = 5 cm
• Semi-perimeter: s = (3 + 4 + 5) / 2 = 6 cm
• Area (Heron's): √[6(6-3)(6-4)(6-5)] = √(6 × 3 × 2 × 1) = √36 = 6 cm²
• Angles (Law of Cosines): A = 36.87°, B = 53.13°, C = 90.00°
• Right triangle: a² + b² = 3² + 4² = 9 + 16 = 25 = 5² = c²

What is the Triangle Calculator?

The Triangle Calculator from SmartCalc solves right-angled and oblique triangles using the Pythagorean theorem, Law of Sines, and Law of Cosines. Enter any three known values (sides and angles) to determine all unknown dimensions, triangle area, perimeter, and inradius/circumradius.

How to Use the Triangle Calculator — Step-by-Step

Follow these quick steps to calculate your results immediately:

  1. 1
    Identify three known values for your triangle (at least one must be a side length).
  2. 2
    Enter these values into the respective fields for sides (a, b, c) and angles (A, B, C).
  3. 3
    Click 'Calculate Triangle' to solve the geometry.
  4. 4
    Review the completed triangle details showing all sides, angles, perimeter, and area.

Pythagoras & Heron's Area Formulas

All calculations are computed based on mathematically rigorous standards and formulas:

Mathematical Formula Pythagoras: a² + b² = c²  |  Heron's Area: A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2

Worked Example

Right triangle with sides a = 3, b = 4:
1. Hypotenuse c = √(3² + 4²) = √(9 + 16) = √25 = 5
2. Area = (3 × 4) / 2 = 6
3. Perimeter = 3 + 4 + 5 = 12

Reference Data & Key Standards

Triangle ClassificationSide RelationshipsAngle Properties
EquilateralAll 3 sides equal (a = b = c)All 3 angles equal to 60°
Isosceles2 sides equal (a = b ≠ c)2 base angles equal
ScaleneAll 3 sides different lengthAll 3 angles different
Right-AngledSatisfies a² + b² = c²One angle is exactly 90°
ObtuseSatisfies a² + b² < c² (c = longest)One angle is greater than 90°

Practical Tips & Best Practices

💡 Pro Tips:

  • The sum of any triangle's interior angles always equals exactly 180°.
  • Triangle Inequality Theorem: The sum of any two sides must always be strictly greater than the third side.
  • For right triangles, the hypotenuse is always the longest side, directly opposite the 90° right angle.

Frequently Asked Questions

Enter three known properties (sides or angles) to calculate all missing side lengths, interior angles, perimeter, and area.

We use Heron's formula if three sides are known, or standard trigonometric area formulas (1/2 * a * b * sin C) for other cases.

The Law of Sines relates sides and angles of any triangle: a/sin(A) = b/sin(B) = c/sin(C).

Key Takeaways
  • Solves right triangles and arbitrary triangles using standard geometric laws.
  • Computes missing sides, area, perimeter, and semi-perimeter.
  • Supports Pythagorean checks and Heron's formula for arbitrary 3-side inputs.
  • Indispensable for carpentry, construction surveying, and trigonometry coursework.