Right Triangle Calculator

About Right Triangle Calculator

Welcome to our Right Triangle Calculator, a professional-grade tool for solving right triangles with precision and clarity. Enter any two known values—whether sides or angles—and instantly compute all missing measurements including sides, angles, area, perimeter, and more with detailed step-by-step solutions and interactive diagrams.

What is a Right Triangle?

A right triangle (or right-angled triangle) is a triangle containing one angle of exactly 90 degrees. This special property makes right triangles fundamental to trigonometry, geometry, and countless real-world applications.

The key components of a right triangle:

• Hypotenuse (c): The side opposite the right angle, always the longest side
• Legs (a and b): The two sides that form the right angle
• Right angle (C): The 90° angle between the two legs
• Acute angles (A and B): The two angles that are less than 90° and sum to 90°

Essential Formulas

Pythagorean Theorem

This fundamental relationship states that the square of the hypotenuse equals the sum of the squares of the two legs.

Trigonometric Ratios

Area and Perimeter

Special Right Triangles

Certain right triangles have special angle relationships with exact trigonometric values:

How to Use This Calculator

1. Identify your known values: Determine which two measurements you have (any combination of sides a, b, c or angles A, B).
2. Enter the values: Input your known values in the corresponding fields. Leave all other fields empty.
3. Click Calculate: The calculator will determine all missing values using the Pythagorean theorem and trigonometric ratios.
4. Review results: Examine the complete solution including all measurements, step-by-step calculations, and the interactive triangle diagram.
5. Verify accuracy: Check the verification section confirming that results satisfy both the Pythagorean theorem and angle sum property.

Solving Methods

When Two Sides Are Known

• Two legs (a and b): Use Pythagorean theorem to find hypotenuse: $c = \sqrt{a^2 + b^2}$. Find angles using inverse tangent: $A = \arctan(a/b)$
• Leg and hypotenuse: Use Pythagorean theorem to find the other leg. Use inverse sine or cosine for angles.

When One Side and One Angle Are Known

• Use the complementary angle relationship: $A + B = 90°$
• Apply the appropriate trigonometric ratio based on which side and angle you know
• Calculate remaining sides using sine, cosine, or tangent

Applications of Right Triangles

Right triangles are essential in numerous fields:

• Architecture & Construction: Calculating roof pitches, stair dimensions, structural supports
• Navigation: Determining distances and bearings, GPS calculations
• Engineering: Force decomposition, mechanical design, electrical circuits
• Surveying: Land measurement, elevation calculations, mapping
• Physics: Vector analysis, projectile motion, inclined planes
• Computer Graphics: 3D rendering, game development, animations
• Astronomy: Celestial distance calculations, parallax measurements

Important Notes

• Exactly two values: Provide exactly two known values—more or fewer will produce an error
• Valid ranges: Angles must be between 0° and 90° (exclusive); all sides must be positive
• Hypotenuse rule: The hypotenuse must be longer than either leg
• Angle format: Enter angles in degrees, not radians
• Consistent units: Use the same unit for all side measurements

Frequently Asked Questions

What is a right triangle?

A right triangle (or right-angled triangle) is a triangle with one angle exactly equal to 90 degrees. The side opposite the right angle is called the hypotenuse and is always the longest side. The other two sides are called legs or catheti.

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b): a² + b² = c². This fundamental relationship allows you to calculate any side if you know the other two.

How do I solve a right triangle with two sides known?

When you know two sides of a right triangle: 1) Use the Pythagorean theorem to find the third side. 2) Use inverse trigonometric functions (arcsin, arccos, arctan) to find the angles. For example, if you know sides a and b, then c = √(a² + b²) and angle A = arctan(a/b).

What are the special right triangles?

The two most common special right triangles are: 1) The 45-45-90 triangle (isosceles right triangle) with sides in ratio 1:1:√2, and 2) The 30-60-90 triangle with sides in ratio 1:√3:2. These triangles have exact trigonometric values and appear frequently in mathematics and engineering.

How do I calculate the area of a right triangle?

The area of a right triangle is calculated using the formula: Area = ½ × base × height. Since the two legs of a right triangle are perpendicular, you simply multiply them together and divide by 2: Area = ½ × a × b, where a and b are the two legs.

Additional Resources

• Right Triangle - Wikipedia
• Pythagorean Theorem - Wikipedia
• Right Triangle - Wolfram MathWorld
• Trigonometric Functions - Wikipedia

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