Free Right Triangle Calculator

Solve any right triangle instantly by entering any two values. Calculate all sides, angles, area, perimeter, altitude to hypotenuse, and trigonometric ratios with step-by-step solutions.

What is a Right Triangle?

A right triangle is a triangle that contains exactly one 90-degree angle, known as the right angle. The two sides that form the right angle are called legs (or catheti), and the side opposite the right angle is called the hypotenuse. The hypotenuse is always the longest side of the right triangle.

Right triangles are among the most important shapes in mathematics. They form the foundation of trigonometry, the Pythagorean theorem, and countless applications in science, engineering, navigation, and everyday life. Every right triangle has three angles that sum to exactly 180 degrees, where one angle is always 90 degrees and the other two are acute angles (less than 90 degrees) that are complementary — meaning they add up to 90 degrees.

Properties of a Right Triangle

Understanding the properties of a right triangle is essential for solving problems accurately:

• One right angle: Exactly one angle measures 90 degrees
• Two acute angles: The remaining two angles are both less than 90 degrees and are complementary (sum to 90 degrees)
• Hypotenuse: The longest side, always opposite the right angle
• Two legs: The sides that form the right angle, sometimes called the base and height
• Angle sum: All three interior angles always add up to 180 degrees
• Pythagorean theorem: The square of the hypotenuse equals the sum of the squares of the legs
• Area: Equals one-half the product of the two legs (Area = ½ × a × b)
• Circumscribed circle: The hypotenuse is the diameter of the circumscribed circle
• Altitude to hypotenuse: The height from the right angle to the hypotenuse equals (a × b) / c

The Pythagorean Theorem

The Pythagorean theorem is the most fundamental relationship in a right triangle. It states that:

a² + b² = c²

Where a and b are the lengths of the two legs, and c is the length of the hypotenuse.

This theorem allows you to find any missing side when you know the other two:

• To find the hypotenuse: c = √(a² + b²)
• To find a leg: a = √(c² - b²) or b = √(c² - a²)

The Pythagorean theorem is named after the ancient Greek mathematician Pythagoras, though the relationship was known to Babylonian and Indian mathematicians centuries earlier. It has been proven in hundreds of different ways and is one of the most verified theorems in mathematics.

Trigonometric Ratios (SOH CAH TOA)

Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. The mnemonic SOH CAH TOA helps you remember these relationships:

Sine (SOH)

sin(θ) = Opposite / Hypotenuse

The sine of an angle equals the length of the opposite side divided by the hypotenuse.

Cosine (CAH)

cos(θ) = Adjacent / Hypotenuse

The cosine of an angle equals the length of the adjacent side divided by the hypotenuse.

Tangent (TOA)

tan(θ) = Opposite / Adjacent

The tangent of an angle equals the length of the opposite side divided by the adjacent side.

For angle A in a right triangle with legs a (opposite to A) and b (adjacent to A) and hypotenuse c:

• sin(A) = a / c
• cos(A) = b / c
• tan(A) = a / b

These ratios are the basis for solving right triangles when you know one side and one angle. Our calculator computes all six trigonometric values (sin, cos, tan) for both acute angles automatically.

Special Right Triangles

30-60-90 Triangle

The 30-60-90 triangle has angles of 30 degrees, 60 degrees, and 90 degrees. The sides always follow the ratio:

• Short leg (opposite 30°): x
• Long leg (opposite 60°): x√3
• Hypotenuse (opposite 90°): 2x

For example, if the short leg is 5, then the long leg is 5√3 ≈ 8.66 and the hypotenuse is 10.

Key trigonometric values:

• sin(30°) = 0.5, cos(30°) = √3/2 ≈ 0.866, tan(30°) = √3/3 ≈ 0.577
• sin(60°) = √3/2 ≈ 0.866, cos(60°) = 0.5, tan(60°) = √3 ≈ 1.732

45-45-90 Triangle (Isosceles Right Triangle)

The 45-45-90 triangle has angles of 45 degrees, 45 degrees, and 90 degrees. It is an isosceles right triangle because the two legs are equal. The sides follow the ratio:

• Both legs: x
• Hypotenuse: x√2

For example, if each leg is 7, the hypotenuse is 7√2 ≈ 9.899.

Key trigonometric values:

• sin(45°) = cos(45°) = √2/2 ≈ 0.707
• tan(45°) = 1

How to Solve a Right Triangle

Solving a right triangle means finding all unknown sides and angles. You need at least two pieces of information (excluding the known 90-degree angle):

Method 1: Two Sides (SS)

When you know any two sides of a right triangle:

1. Know legs a and b: Use the Pythagorean theorem to find c, then use inverse trigonometric functions for the angles
2. Know one leg and hypotenuse: Use the Pythagorean theorem to find the missing leg, then calculate angles
3. All side combinations work as long as the hypotenuse is the longest side

Method 2: One Side and One Angle (SA)

When you know one side and one acute angle:

1. Find the other acute angle (subtract from 90 degrees)
2. Use sine, cosine, or tangent to calculate the remaining sides
3. Choose the trigonometric function that directly involves the known side

Method 3: Area and One Angle

When you know the area and one angle:

1. Use the area formula combined with trigonometric relationships
2. Express one leg in terms of the other using the tangent of the known angle
3. Substitute into the area formula and solve

Area of a Right Triangle

The area of a right triangle is calculated using the formula:

Area = ½ × a × b

Where a and b are the lengths of the two legs (the sides that form the right angle). This formula works because the legs of a right triangle serve as the base and height, making the calculation straightforward.

You can also express the area in terms of the hypotenuse:

Area = ½ × c × h

Where h is the altitude (height) from the right angle vertex to the hypotenuse.

Perimeter of a Right Triangle

The perimeter is simply the sum of all three sides:

Perimeter = a + b + c

Once you have found all three sides using the methods described above, adding them together gives you the perimeter. This is useful in construction, fencing, and framing applications.

Altitude to the Hypotenuse

The altitude from the right angle vertex to the hypotenuse is a useful measurement in geometry. It is calculated as:

h = (a × b) / c

This altitude divides the original right triangle into two smaller right triangles that are similar to each other and to the original triangle. This property leads to several important geometric relationships used in advanced proofs and constructions.

Applications of Right Triangles

Construction and Carpentry

Builders and carpenters use right triangles constantly. The 3-4-5 method for checking square corners, calculating rafter lengths, determining roof pitches, and laying out foundations all rely on right triangle geometry. Every framed wall, every roof truss, and every deck involves right triangle calculations.

Navigation and GPS

Modern GPS systems use right triangle trigonometry to calculate distances and positions. When determining how far you are from a destination, the system often breaks the displacement into north-south and east-west components, then uses the Pythagorean theorem to find the straight-line distance.

Physics and Engineering

Engineers use right triangles to resolve force vectors, calculate projectile trajectories, determine structural loads, and design mechanical systems. In electrical engineering, impedance calculations in AC circuits use right triangle relationships between resistance, reactance, and impedance.

Architecture

Architects rely on right triangle calculations for roof slopes, stair dimensions, ADA-compliant ramps, and structural analysis. Every pitched roof creates a right triangle where the rise, run, and rafter length must be precisely calculated.

Computer Graphics and Game Development

Screen coordinate systems, collision detection, camera positioning, and character movement all involve right triangle calculations. The distance between two pixels on screen is calculated using the Pythagorean theorem applied to their coordinate differences.

Surveying and Land Measurement

Surveyors use right triangles to measure distances that cannot be directly accessed. By setting up known angles and measuring accessible sides, they can calculate distances across rivers, canyons, and other obstacles.

Astronomy

Astronomers use right triangle trigonometry to calculate distances to stars and planets using parallax measurements. The angular position of celestial objects and their apparent motion create right triangles on astronomical scales.

Common Pythagorean Triples

Pythagorean triples are sets of three positive integers that satisfy the equation a² + b² = c². Knowing these common triples helps you quickly verify calculations:

Any integer multiple of a Pythagorean triple is also a valid triple. For example, multiplying 3-4-5 by 7 gives 21-28-35.

Frequently Asked Questions

What is a right triangle?

A right triangle is a triangle that has exactly one angle measuring 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse and is always the longest side. The other two sides are called legs.

How do I find the hypotenuse of a right triangle?

Use the Pythagorean theorem: c = √(a² + b²). Square both legs, add them together, and take the square root of the result. For example, if a = 3 and b = 4, then c = √(9 + 16) = √25 = 5.

How do I find a missing angle in a right triangle?

Since the two acute angles in a right triangle are complementary (they add up to 90 degrees), if you know one acute angle, subtract it from 90 to find the other. You can also use inverse trigonometric functions: A = arcsin(a/c), A = arccos(b/c), or A = arctan(a/b).

What is the formula for the area of a right triangle?

The area of a right triangle equals half the product of its two legs: Area = ½ × a × b. This works because the legs are perpendicular, so one leg serves as the base and the other as the height.

What is SOH CAH TOA?

SOH CAH TOA is a mnemonic for remembering trigonometric ratios in right triangles. SOH: Sine = Opposite / Hypotenuse. CAH: Cosine = Adjacent / Hypotenuse. TOA: Tangent = Opposite / Adjacent. These ratios relate angles to side lengths.

How do I calculate the altitude to the hypotenuse?

The altitude from the right angle vertex to the hypotenuse is calculated using the formula: h = (a × b) / c, where a and b are the legs and c is the hypotenuse. This altitude creates two smaller similar triangles within the original.

What is a 45-45-90 triangle?

A 45-45-90 triangle is an isosceles right triangle where both acute angles measure 45 degrees and the two legs are equal in length. The hypotenuse is always √2 times the length of each leg. It is one of the two special right triangles.

What is a 30-60-90 triangle?

A 30-60-90 triangle has angles of 30, 60, and 90 degrees. The sides follow a fixed ratio of 1 : √3 : 2, where the shortest side (opposite the 30° angle) is x, the longer leg (opposite the 60° angle) is x√3, and the hypotenuse is 2x.

Can I solve a right triangle with only one known value?

No, you need at least two pieces of information (excluding the known 90-degree angle) to solve a right triangle. This can be any combination of two sides, one side and one angle, or the area plus one other measurement.

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides. It is written as a² + b² = c² and only applies to right triangles.

Why Use Our Right Triangle Calculator

• Free and instant — No sign-up required, results calculated in real-time
• Multiple input modes — Enter any 2 of 6 possible values to solve
• Complete solution — All sides, angles, area, perimeter, altitude, and trig ratios
• Visual SVG diagram — Color-coded right triangle with labeled sides and angle markers
• Step-by-step solutions — See every calculation step clearly explained
• Trigonometric ratios — Full sin, cos, tan values for both acute angles
• Quick examples — Load common triangles with one click
• Mobile friendly — Works perfectly on phones, tablets, and desktops
• Accurate results — High-precision calculations with proper mathematical functions