Pythagorean Theorem Angle Calculator

Calculate Angles from Side Lengths

Enter the side lengths to find all angles in the triangle

Calculation Method:

Side a (opposite to angle A):

Side b (opposite to angle B):

Side c (hypotenuse):

1. Enter a known angle (A or B): deg

2. Enter its corresponding side length:

3. Specify which side this is:

Angle Results:

Trigonometric Formulas

Sine: sin(A) = opposite / hypotenuse = a / c

Cosine: cos(A) = adjacent / hypotenuse = b / c

Tangent: tan(A) = opposite / adjacent = a / b

Inverse: A = arcsin(a/c) = arccos(b/c) = arctan(a/b)

Quick Examples

3-4-5 Triangle:
Sides: a=3, b=4, c=5
Angle A = arcsin(3/5) ≈ 36.87°
Angle B = arcsin(4/5) ≈ 53.13°

5-12-13 Triangle:
Sides: a=5, b=12, c=13
Angle A = arcsin(5/13) ≈ 22.62°
Angle B = arcsin(12/13) ≈ 67.38°

45° Right Triangle:
Equal legs: a=b=1, c=√2
Both angles A and B = 45°
Perfect square triangle

How to Calculate Angles in a Right Triangle

Understanding the Basics: SOH CAH TOA

To find the angles of a right triangle, we use trigonometry. The mnemonic SOH CAH TOA is a simple way to remember the main trigonometric ratios:

SOH: Sine(θ) = Opposite / Hypotenuse
CAH: Cosine(θ) = Adjacent / Hypotenuse
TOA: Tangent(θ) = Opposite / Adjacent

Our calculator uses these fundamental formulas to solve for unknown angles and sides.

Finding Angles from Side Lengths

When you know the lengths of at least two sides, you can use the inverse trigonometric functions (also known as arc-functions) to find the angle.

If you know the Opposite and Hypotenuse, use arcsin(a/c).
If you know the Adjacent and Hypotenuse, use arccos(b/c).
If you know the Opposite and Adjacent, use arctan(a/b).

This is what our "From Side Lengths" method does for you automatically.

Angle Calculator Features

📐

Multiple Methods

Calculate angles from side lengths or find missing sides from known angles

🧮

Sin Cos Tan Functions

Uses trigonometric functions with inverse calculations (arcsin, arccos, arctan)

✅

Validation

Automatically checks if the triangle is valid and forms a right triangle

📊

Detailed Steps

See the complete calculation process with trigonometric formulas

Common Applications

🏗️ Construction & Engineering

Calculate roof angles, ramp slopes, and structural angles for building projects.

📐 Surveying & Navigation

Determine angles for land surveying, GPS calculations, and navigation systems.

🎯 Physics & Science

Analyze force vectors, projectile motion, and wave calculations in physics.

🎮 Game Development

Calculate angles for character aiming, object rotation, and camera positioning in 2D and 3D games.

📚 Education & Learning

Perfect for students learning trigonometry, geometry, and applied mathematics.

Frequently Asked Questions (FAQ)

How do you find the angle of a right triangle?

If you know two sides of the triangle, you can use inverse trigonometric functions to find the angles:

α = arcsin(opposite/hypotenuse) = arcsin(a/c)
α = arccos(adjacent/hypotenuse) = arccos(b/c)
α = arctan(opposite/adjacent) = arctan(a/b)

Can you solve a right triangle with only one side?

No, it's impossible to solve a right triangle if you only know the length of one side. You need at least two pieces of information: either two side lengths, or one side length and one of the non-right angles.

What is the difference between sine, cosine, and tangent?

They are the three main trigonometric ratios that relate the angles of a right triangle to the ratios of its side lengths.

Sine (sin) relates the angle to the ratio of the opposite side and the hypotenuse.
Cosine (cos) relates the angle to the ratio of the adjacent side and the hypotenuse.
Tangent (tan) relates the angle to the ratio of the opposite and adjacent sides.