Right Triangle Calculator

Solve for missing sides, angles, area, perimeter, and more. Select a mode to begin.

Legs (a, b)

Enter the two legs to find the hypotenuse and other properties.

Side a (Opposite)

Side b (Adjacent)

Leg & Hypotenuse

Enter one leg and the hypotenuse to find the missing leg.

Side a (Known Leg)

Side c (Hypotenuse)

Side & Angle

Enter one side and one acute angle (α or β).

Known Side Side a (Opposite α) Side b (Adjacent α) Side c (Hypotenuse)

Known Side Length

Known Angle α (opposite a)

Options

Units Meters (m) Centimeters (cm) Millimeters (mm) Kilometers (km) Inches (in) Feet (ft) Yards (yd) Miles (mi)

Angle Units Degrees (°) Radians (rad)

Precision 0 1 2 3 4 5 6

Data stays on this device. For engineering or safety-critical tasks, consult a qualified professional.

Results

Triangle Properties

What Is a Right Triangle?

A right triangle, also known as a right-angled triangle, is a triangle in which one angle is exactly 90 degrees. This 90-degree angle is called the right angle. The side opposite the right angle is the longest side and is called the hypotenuse. The other two sides are called legs (or sometimes cathetus). The relationship between the sides of a right triangle is fundamental to geometry and trigonometry and is described by the famous Pythagorean theorem.

The Pythagorean Theorem: a² + b² = c²

The cornerstone of right triangle mathematics is the Pythagorean theorem. It states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the two legs (a and b). The formula is written as:

a² + b² = c²

This theorem allows you to find the length of a missing side if you know the lengths of the other two sides. For example, to find the hypotenuse, you can rearrange the formula to c = √(a² + b²). To find a missing leg (e.g., 'a'), you would use a = √(c² - b²). Our calculator automates these steps for you, providing a quick and error-free solution.

How to Solve a Right Triangle

To "solve" a right triangle means to find all its unknown sides and angles. You can do this with a minimal amount of information. This calculator supports the most common scenarios:

• Given two legs (a and b): First, use the Pythagorean theorem to find the hypotenuse (c). Then, use trigonometric functions like arctangent (α = arctan(a/b)) to find one of the acute angles. The third angle is simply 90 minus the one you just found, as the sum of angles in any triangle is 180 degrees.
• Given one leg and the hypotenuse (a and c): Use the Pythagorean theorem to find the other leg (b = √(c² - a²)). Then, find the angles using functions like arcsin (α = arcsin(a/c)).
• Given one side and one acute angle (a and α): You can find the other sides using trigonometric ratios. For example, since sin(α) = a/c, the hypotenuse is c = a / sin(α). And since tan(α) = a/b, the other leg is b = a / tan(α). The other acute angle is β = 90° - α.

Trigonometric Ratios Explained (SOH-CAH-TOA)

Trigonometry is built on the relationships between the angles and side ratios of a right triangle. The three primary ratios are Sine, Cosine, and Tangent, often remembered by the mnemonic "SOH-CAH-TOA".

• Sine (sin): Opposite / Hypotenuse. sin(α) = a/c
• Cosine (cos): Adjacent / Hypotenuse. cos(α) = b/c
• Tangent (tan): Opposite / Adjacent. tan(α) = a/b

There are also three reciprocal ratios: Cosecant (csc = 1/sin), Secant (sec = 1/cos), and Cotangent (cot = 1/tan). These ratios are crucial for solving triangles when you have a mix of sides and angles.

Practical Applications

Right triangles are not just a concept for math class; they are everywhere in the real world:

• Construction & Architecture: Ensuring walls are perpendicular (forming a right angle with the floor), calculating roof pitches, and designing stable support structures like trusses.
• Navigation: GPS systems and historical navigation methods use triangulation, which breaks down complex paths into a series of triangles, to pinpoint locations.
• Surveying: Land surveyors use theodolites to measure angles and distances, applying trigonometry to calculate the dimensions and areas of properties.
• Computer Graphics: 3D models are composed of millions of tiny triangles (a "mesh"). Right triangle math is essential for calculating lighting, shadows, and perspective.
• Physics: Resolving force vectors into their horizontal and vertical components is a direct application of right triangle principles.

Frequently Asked Questions

Why does the hypotenuse have to be the longest side?

In the formula c² = a² + b², since a² and b² are always positive, c² must be greater than both a² and b². Therefore, c must be greater than both a and b.

What is the difference between degrees and radians?

They are two different units for measuring angles. A full circle is 360 degrees or 2π radians. Degrees are common in everyday life, while radians are preferred in higher-level mathematics and physics for their more direct relationship with the radius of a circle.

What are the inradius and circumradius?

The inradius is the radius of the largest circle that can be inscribed within the triangle (the incircle). The circumradius is the radius of the circle that passes through all three vertices of the triangle (the circumcircle). For a right triangle, the circumradius is always half the length of the hypotenuse.

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Disclaimer: This calculator is a tool for educational purposes and quick estimations. It should not be used for safety-critical, engineering, or construction applications where precise calculations are required. Always verify results with a qualified professional for such tasks.