Right Triangle Calculator

Calculate all properties of a right triangle with any two known values. Get instant results for sides, angles, area, perimeter, and height.

📐 Right Triangle Calculator

Triangle Diagram

Right angle at vertex C (90°)

What is a Right Triangle Calculator?

A right triangle calculator is a powerful online tool that solves all properties of a right triangle when you provide any two independent values. Whether you know two sides, one side and one angle, or other combinations, the calculator instantly computes all remaining measurements including the third side, both acute angles, area, perimeter, and height.

Right triangles are fundamental geometric shapes characterized by one 90-degree angle. They appear everywhere in mathematics, engineering, construction, and daily life. Understanding their properties is essential for solving real-world problems involving distances, heights, slopes, and angles.

Our calculator uses the Pythagorean theorem (c² = a² + b²) and trigonometric functions (sine, cosine, tangent) to accurately compute all triangle properties. It supports both degrees and radians for angle measurements and provides results with high precision for professional and educational use.

How to Use the Right Triangle Calculator

Enter any two known values from the available fields: sides (a, b, c), height (h), angles (α, β), area, or perimeter.
For angles, select whether you're entering degrees or radians using the dropdown menu next to each angle field.
Click the 'Calculate' button to compute all remaining properties of the right triangle.
Review the comprehensive results showing all sides, angles (in degrees), area, perimeter, and height with four decimal places of precision.
Use the 'Clear' button to reset all fields and start a new calculation.

Latest Insights on Right Triangle Calculations

Core Mathematical Formulas

Right triangle calculators rely on the Pythagorean theorem (c² = a² + b²) for calculating the hypotenuse and basic trigonometric ratios (sin, cos, tan) for determining unknown sides and angles. The area formula A = (a × b) / 2 uses the two legs that form the right angle. These fundamental relationships make right triangle calculations both reliable and versatile.

Automation Benefits

These calculators automate multi-step geometric and trigonometric calculations, significantly reducing arithmetic errors and providing instant results. They're invaluable for education, construction (calculating rafters and slopes), surveying, navigation, and design tasks that depend on right-triangle geometry. The automation saves time and ensures accuracy in professional applications.

Modern Calculator Features

Contemporary right triangle calculators offer unit selection (cm, m, in, ft), angle input in both degrees and radians, and often include step-by-step solutions or visual diagrams. These features make them suitable for both quick calculations and educational purposes, helping users understand how right-triangle trigonometry works in practice.

Best Practices for Accurate Results

Always confirm which side is the hypotenuse (opposite the 90° angle), ensure your calculator's angle mode matches your input (degrees vs radians), use consistent units for all sides, and provide only valid pairs of inputs (two sides or one side plus one acute angle) to uniquely determine the triangle. These practices ensure reliable and meaningful results.

Special Right Triangles

Calculators can exploit known side ratios for special right triangles like 30°-60°-90° (ratio 1:√3:2) and 45°-45°-90° (ratio 1:1:√2) to speed up calculations and verify results. These special cases are particularly useful in education and quick design estimates where these common angles frequently appear.

Understanding Right Triangle Calculations

The Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of squares of the other two sides: c² = a² + b². This fundamental relationship allows you to find any side when the other two are known. It's one of the most important theorems in mathematics with countless practical applications.

Trigonometric Relationships

The six trigonometric functions relate the angles and sides of right triangles. For an angle θ: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. These ratios allow you to find unknown sides when you know one side and one angle, or find angles when you know two sides.

Area and Perimeter Formulas

The area of a right triangle is calculated as A = (a × b) / 2, where a and b are the two legs forming the right angle. The perimeter is simply P = a + b + c, the sum of all three sides. The height to the hypotenuse can be found using h = (a × b) / c, which is useful in various geometric constructions.

Real-World Applications

Construction: Calculating roof pitches, rafter lengths, and stair dimensions
Surveying: Determining distances and elevations using angle measurements
Navigation: Computing travel distances and bearings
Engineering: Designing structures with precise angle and length requirements
Physics: Analyzing force vectors and projectile motion

Frequently Asked Questions

What is the minimum information needed to solve a right triangle?

You need at least two independent pieces of information (excluding the right angle). This can be two sides, one side and one acute angle, or in some cases, the area plus one side. The calculator will then determine all other properties.

Can I use this calculator for non-right triangles?

No, this calculator is specifically designed for right triangles (triangles with one 90° angle). For arbitrary triangles, you would need to use the Law of Sines or Law of Cosines, which are different calculation methods.

What's the difference between degrees and radians?

Degrees and radians are two units for measuring angles. A full circle is 360 degrees or 2π radians. To convert: degrees = radians × (180/π), and radians = degrees × (π/180). Most everyday applications use degrees, while advanced mathematics often uses radians.

Why does my calculation show an error?

Common reasons include: entering fewer than two values, providing values that don't form a valid right triangle (e.g., hypotenuse shorter than a leg), entering negative numbers, or providing incompatible combinations. Ensure your inputs represent a physically possible right triangle.

How accurate are the calculator results?

The calculator provides results with four decimal places of precision, which is suitable for most practical applications. However, remember that real-world measurements have tolerances, so always consider measurement accuracy and rounding when applying results to construction or engineering projects.

What are the special right triangles?

The two most common special right triangles are the 30°-60°-90° triangle (with side ratio 1:√3:2) and the 45°-45°-90° triangle (with side ratio 1:1:√2). These appear frequently in geometry and have predictable side relationships that make calculations easier.

Right Triangle Calculator