Right Triangle Calculator

Calculate the missing sides, angles, area, and perimeter of a right-angled triangle by providing any two known values (at least one side).

Enter any two values to solve the triangle.

Examples

Click on an example to load the data into the calculator.

Two Legs

a_and_b

Given the two legs (side a and side b) of the triangle.

sideA: 3

sideB: 4

Leg and Opposite Angle

a_and_A

Given one leg (side a) and its opposite angle (angle A).

sideA: 5

angleA: 30

Hypotenuse and Angle

c_and_B

Given the hypotenuse (c) and one adjacent angle (angle B).

hypotenuseC: 10

angleB: 60

Leg and Hypotenuse

b_and_c

Given one leg (side b) and the hypotenuse (c).

sideB: 12

hypotenuseC: 13

Other Titles
Understanding the Right Triangle Calculator: A Comprehensive Guide
Master the geometry of right triangles, from the Pythagorean theorem to trigonometric functions, and learn how to solve for missing sides and angles with ease.

What is a Right Triangle?

  • Defining a right triangle and its components.
  • The fundamental role of the Pythagorean theorem.
  • Introduction to trigonometric ratios (SOH-CAH-TOA).
A right triangle, also known as a right-angled triangle, is a triangle in which one angle is exactly 90 degrees. This right angle is the defining feature. The side opposite the right angle is called the hypotenuse, which is always the longest side. The other two sides are called the legs (or catheti).
Pythagorean Theorem
The relationship between the sides is famously described by the Pythagorean theorem: a² + b² = c². In this formula, 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse. This theorem allows you to find the length of a missing side if you know the lengths of the other two.
Trigonometric Ratios
Trigonometry provides the tools to relate angles and side lengths. The three primary ratios are Sine (sin), Cosine (cos), and Tangent (tan):
Sine (A) = Opposite / Hypotenuse (a/c)
Cosine (A) = Adjacent / Hypotenuse (b/c)
Tangent (A) = Opposite / Adjacent (a/b)
The mnemonic SOH-CAH-TOA is often used to remember these relationships.

Key Properties

  • Sum of angles is 180° (A + B + 90° = 180°).
  • The two acute angles are complementary (A + B = 90°).
  • The famous 3-4-5 triangle is a right triangle where 3² + 4² = 5².

Step-by-Step Guide to Using the Calculator

  • How to input known values correctly.
  • Understanding the calculated results.
  • Using the examples to get started quickly.
Our calculator is designed to be intuitive. Follow these simple steps to solve your right triangle problems:
1. Identify Your Knowns
Look at your triangle problem and determine which two values you have. You must know at least one side length. The possible inputs are Side a, Side b, Hypotenuse c, Angle A, and Angle B.
2. Enter the Values
Enter your two known values into their corresponding fields in the calculator. Leave the other three fields blank. For example, if you know Side a is 5 and Angle A is 30°, you would fill only those two inputs.
3. Calculate and Interpret
Click the 'Calculate' button. The calculator will instantly display all missing values: the three sides, the two acute angles (in degrees), the triangle's area, and its perimeter.

Input Scenarios Handled

  • Two Legs (a, b)
  • Leg and Hypotenuse (e.g., a, c)
  • Leg and an Angle (e.g., a, A or a, B)

Real-World Applications of Right Triangles

  • Applications in architecture and construction.
  • How navigation and surveying rely on trigonometry.
  • Uses in physics, engineering, and computer graphics.
The principles of right triangles are not just academic; they are the foundation for many practical applications in science, technology, and everyday life.
Architecture and Construction
Architects and builders use right triangles constantly to ensure corners are perfectly square, to calculate the pitch of a roof, design staircases, or determine the length of support beams.
Navigation and Surveying
Surveyors use triangulation, a method based on triangle properties, to determine distances between points. GPS and other navigation systems use similar principles to pinpoint locations.
Physics and Engineering
In physics, forces and velocities are often represented as vectors. Right triangles are used to break these vectors down into their horizontal and vertical components, which is essential for analyzing motion and forces.

Practical Examples

  • Calculating the height of a building by measuring the distance from its base and the angle to its top.
  • Finding the shortest distance (a straight line, the hypotenuse) to a location.
  • Designing a wheelchair ramp with a safe and accessible slope (angle).

Common Misconceptions and Correct Methods

  • Confusing sine, cosine, and tangent.
  • Errors in applying the Pythagorean theorem.
  • Mistakes with angle units (degrees vs. radians).
While powerful, trigonometric calculations can be prone to common errors. Understanding these pitfalls is key to accurate problem-solving.
Mixing up SOH-CAH-TOA
Misconception: Using the wrong trigonometric ratio for a given pair of sides. Correction: Always double-check the relationship: Sine is Opposite/Hypotenuse, Cosine is Adjacent/Hypotenuse, and Tangent is Opposite/Adjacent. Labeling the sides of your triangle first can help prevent this.
Pythagorean Theorem Errors
Misconception: Incorrectly setting up the theorem, such as a² + c² = b². Correction: The hypotenuse (c) must always be by itself on one side of the equation (a² + b² = c²). When solving for a leg, the formula becomes a² = c² - b².
Angle Units
Misconception: Using radians in a calculator set to degrees, or vice-versa. Correction: Our calculator exclusively uses degrees for all angle inputs and outputs, which is the most common unit for geometry problems. Be mindful of this when working with other tools.

Key Takeaways

  • Always identify the hypotenuse first.
  • Use inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) to find angles.
  • Ensure your inputs are positive values as length and angle are physical quantities.

Mathematical Derivations and Formulas

  • The formulas used for each calculation scenario.
  • Derivation of area and perimeter.
  • How inverse trigonometric functions find angles.
The calculator solves for unknowns based on a set of logical rules and core formulas derived from geometry and trigonometry.
Core Formulas
Pythagorean Theorem: c = sqrt(a² + b²)
Sine: sin(A) = a/c
Cosine: cos(A) = b/c
Tangent: tan(A) = a/b
Angle Sum: A + B = 90°
Derived Formulas
Depending on the inputs, the calculator combines these formulas. For example, if 'a' and 'A' are known:
B = 90 - A
c = a / sin(A)
b = a / tan(A)
Area and Perimeter
Area = (1/2) a b
Perimeter = a + b + c

Example Derivation (Given a=3, b=4)

  • `c = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5`
  • `A = atan(3/4) ≈ 36.87°`
  • `B = atan(4/3) ≈ 53.13°`