Right Triangle Solver

Enter any two values to solve the triangle.

Provide at least two values to calculate the remaining properties of the right-angled triangle, including sides, angles, area, and perimeter.

degrees
degrees
Practical Examples

Click on an example to see how the calculator works with different inputs.

Given Two Legs (a, b)

aa-bb

Calculate hypotenuse, angles, area, and perimeter from two sides.

Inputs: sideA: 3, sideB: 4

Given Leg and Hypotenuse (a, c)

ac-A

Calculate the other leg, angles, area, and perimeter.

Inputs: sideA: 5, sideC: 13

Given Leg and Angle (a, A)

a-A

Calculate all sides and the other angle from one side and its opposite angle.

Inputs: sideA: 7, angleA: 30

Given Hypotenuse and Angle (c, B)

c-B

Calculate legs and the other angle from the hypotenuse and one angle.

Inputs: sideC: 10, angleB: 60

Other Titles
Understanding the Right Triangle Calculator: A Comprehensive Guide
Explore the principles behind solving right triangles, from the Pythagorean theorem to trigonometric functions, and their real-world significance.

What is a Right Triangle?

  • Defining Characteristics
  • Key Terminology
  • The 90-Degree Angle Rule
A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This unique property makes it a cornerstone of geometry and trigonometry. The sides and angles of a right triangle are intrinsically linked by simple yet powerful mathematical rules.
Key Components
Every right triangle has three sides: two 'legs' (sides a and b) and a 'hypotenuse' (side c). The legs are the two sides that form the right angle. The hypotenuse is the longest side and is always opposite the right angle. The other two angles (A and B) are always acute, meaning they are less than 90 degrees, and their sum is always 90 degrees.

Fundamental Properties

  • a² + b² = c² (Pythagorean Theorem)
  • A + B = 90° (Sum of acute angles)
  • Area = (1/2) * a * b
  • Perimeter = a + b + c

Step-by-Step Guide to Using the Calculator

  • Inputting Your Values
  • The Calculation Process
  • Interpreting the Results
Our Right Triangle Calculator is designed for ease of use. By providing just two pieces of information, you can solve for all the unknown properties of the triangle. The tool automatically determines which formulas to apply based on your inputs.
How It Works
1. Enter Knowns: Fill in any two of the five input fields: Side a, Side b, Hypotenuse c, Angle A, or Angle B.
2. Calculate: Click the 'Calculate' button. The tool validates the input to ensure it can form a valid right triangle.
3. View Results: The calculator instantly displays all properties: the lengths of all three sides, the measures of all three angles (including the 90° angle), the total area, and the perimeter.
The tool handles various input combinations, such as two legs, a leg and a hypotenuse, or a side and an angle.

Example Scenarios

  • Input 'Side a = 3' and 'Side b = 4' to find Hypotenuse c = 5.
  • Input 'Hypotenuse c = 10' and 'Angle A = 30°' to find Side a = 5.

Real-World Applications of Right Triangles

  • Architecture and Construction
  • Navigation and Surveying
  • Physics and Engineering
The principles of right triangles are not just academic; they are applied every day in numerous professional fields to solve practical problems.
Practical Uses
Construction: Architects and builders use right triangles to design stable structures, calculate roof pitches, and ensure that corners are perfectly square (90 degrees).
Navigation: Pilots and sailors use triangulation, which relies on right triangle principles, to determine their position relative to fixed points.
Surveying: Land surveyors use trigonometry to measure distances and elevations that are otherwise inaccessible, like the height of a mountain or the width of a river.
Physics: Engineers use right triangles to analyze vectors, such as resolving forces and velocities into their horizontal and vertical components.

Application Examples

  • Calculating the height of a flagpole by measuring the distance from its base and the angle of elevation to the top.
  • Determining the length of a ramp needed to reach a certain height at a specific incline.

Common Misconceptions and Correct Methods

  • Pythagorean Theorem Misuse
  • Angle Unit Confusion
  • Hypotenuse Identification
While right triangle calculations are straightforward, a few common mistakes can lead to incorrect results. Understanding these pitfalls is key to accurate problem-solving.
Points of Caution
Pythagorean Theorem: A common error is misapplying the formula a² + b² = c². It only works for right triangles, and 'c' must always be the hypotenuse. You cannot use it to find a leg if you mix up the variables.
Degrees vs. Radians: Most scientific calculators and programming languages use radians for trigonometric functions by default. Ensure your angles are in degrees when using this tool, or convert them if necessary. Our calculator exclusively uses degrees.
Identifying the Hypotenuse: Always confirm that the hypotenuse (if given or calculated) is the longest side. If a leg is longer than the hypotenuse, the triangle is impossible.

Verification Checks

  • Always ensure side c > side a and side c > side b.
  • Verify that the calculated angles A and B sum to 90 degrees.

Mathematical Derivations and Formulas

  • The Pythagorean Theorem
  • SOHCAHTOA Trigonometric Ratios
  • Calculating Area and Perimeter
The calculations performed by this tool are based on fundamental, time-tested mathematical formulas that govern the relationships within a right triangle.
Core Formulas
Pythagorean Theorem: Used to relate the three sides. c = √(a² + b²)
Trigonometric Ratios (SOHCAHTOA): Used to relate sides and angles.
- sin(θ) = Opposite / Hypotenuse (e.g., sin(A) = a / c)
- cos(θ) = Adjacent / Hypotenuse (e.g., cos(A) = b / c)
- tan(θ) = Opposite / Adjacent (e.g., tan(A) = a / b)
Inverse Trigonometry: Used to find angles from side ratios (e.g., A = arcsin(a/c)).
Area & Perimeter: Area = 0.5 * a * b and Perimeter = a + b + c.

Formula Applications

  • If a=3, b=4, then c = √(3²+4²) = √25 = 5.
  • If a=5, c=10, then sin(A) = 5/10 = 0.5, so A = arcsin(0.5) = 30°.