Right Triangle Calculator: Pythagorean theorem and trigonometry

A right triangle has a 90° angle and is the foundation of trigonometry. This calculator uses the Pythagorean theorem and sine, cosine and tangent functions to find unknown sides and angles.

Pythagorean theorem and trigonometry

In a right triangle with legs a, b and hypotenuse c:

• Pythagoras: a² + b² = c²
• Sine: sin(θ) = opposite / hypotenuse
• Cosine: cos(θ) = adjacent / hypotenuse
• Tangent: tan(θ) = opposite / adjacent

Knowing two sides or one side and one angle, you can determine all remaining values of the triangle.

Example 1: finding the hypotenuse

Problem: A right triangle has legs a = 3 cm and b = 4 cm.

1. Apply Pythagoras:
   • c² = 3² + 4² = 9 + 16 = 25.
2. Hypotenuse:
   • c = √25 = 5 cm.

Answer: c = 5 cm (the famous 3-4-5 triangle).

Example 2: finding an angle

Problem: A triangle has opposite leg = 5 cm and adjacent leg = 12 cm.

1. Hypotenuse:
   • c = √(5² + 12²) = √(25 + 144) = √169 = 13 cm.
2. Angle θ:
   • tan(θ) = 5/12 ≈ 0.4167.
   • θ = arctan(0.4167) ≈ 22.62°.

Answer: c = 13 cm, θ ≈ 22.62°.

Common uses of right triangles

• Computing inaccessible distances and heights in surveying.
• Determining the incline of ramps, stairs and roofs.
• Solving navigation and orientation problems.
• Analyzing forces and components in physics and engineering.
• Computing diagonal distances on screens and monitors.
• Foundation of applied trigonometry in architecture.

Common mistakes with right triangles

• Confusing the opposite leg with the adjacent leg relative to the reference angle.
• Using degrees instead of radians (or vice versa) on the calculator.
• Applying Pythagoras to triangles that are not right triangles.
• Forgetting that the hypotenuse is always the longest side.

Pro tip

Memorize common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and special angles (30°-60°-90° and 45°-45°-90°). They will save you calculations in many practical problems.