Right Angle Triangle Calculator
This calculator helps you find the missing sides and angles of a right-angle triangle. A right-angle triangle is a triangle in which one angle is exactly 90 degrees. The sides adjacent to the right angle are called "legs" (Side A and Side B), and the side opposite the right angle is called the "hypotenuse" (Hypotenuse C).
To use the calculator, enter any two known values (sides or acute angles). The calculator will then determine the remaining sides and angles using the Pythagorean theorem and trigonometric ratios.
Understanding Right Angle Triangles
A right-angle triangle is fundamental in geometry and trigonometry. It consists of three sides and three angles. One angle is always 90 degrees (the right angle), and the other two angles are acute (less than 90 degrees). The sum of all angles in any triangle is always 180 degrees, so in a right-angle triangle, the two acute angles must sum up to 90 degrees.
Key Components:
- Legs (Side A and Side B): These are the two shorter sides that form the 90-degree angle.
- Hypotenuse (Hypotenuse C): This is the longest side of the triangle, always opposite the right angle.
- Angle Alpha (α): The acute angle opposite Side A.
- Angle Beta (β): The acute angle opposite Side B.
- Angle Gamma (γ): The right angle, always 90 degrees.
The Pythagorean Theorem
The most famous relationship in a right-angle triangle is the Pythagorean theorem, which states: a² + b² = c². Where '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 any side if the other two sides are known.
Trigonometric Ratios (SOH CAH TOA)
Trigonometry provides relationships between the angles and sides of a right-angle triangle. These are often remembered by the acronym SOH CAH TOA:
- SOH: Sin(Angle) = Opposite / Hypotenuse
- CAH: Cos(Angle) = Adjacent / Hypotenuse
- TOA: Tan(Angle) = Opposite / Adjacent
Using these ratios, if you know one side and one acute angle, or two sides, you can find all other missing values.
How to Use This Calculator
- Identify Known Values: Look at your right-angle triangle problem and determine which sides (A, B, C) or angles (Alpha, Beta) you already know.
- Enter Values: Input your known values into the corresponding fields in the calculator. You must enter at least two values.
- Click "Calculate": The calculator will process your inputs and display the missing sides and angles.
- Review Results: The results will show all sides and angles, rounded to four decimal places.
- "Clear" Button: Use this to reset all input fields and start a new calculation.
Examples
Here are a few common scenarios:
- Example 1: Given two legs (Side A and Side B)
- Input: Side A = 3, Side B = 4
- Output: Hypotenuse C = 5, Angle Alpha = 36.8699 degrees, Angle Beta = 53.1301 degrees
- Example 2: Given one leg and the hypotenuse (Side A and Hypotenuse C)
- Input: Side A = 6, Hypotenuse C = 10
- Output: Side B = 8, Angle Alpha = 36.8699 degrees, Angle Beta = 53.1301 degrees
- Example 3: Given one leg and an acute angle (Side B and Angle Alpha)
- Input: Side B = 7, Angle Alpha = 45 degrees
- Output: Side A = 7, Hypotenuse C = 9.8995, Angle Beta = 45 degrees
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function solveTriangle() {
var a_input = parseFloat(document.getElementById('sideA').value);
var b_input = parseFloat(document.getElementById('sideB').value);
var c_input = parseFloat(document.getElementById('hypotenuseC').value);
var alpha_input = parseFloat(document.getElementById('angleAlpha').value);
var beta_input = parseFloat(document.getElementById('angleBeta').value);
var knownCount = 0;
if (!isNaN(a_input)) knownCount++;
if (!isNaN(b_input)) knownCount++;
if (!isNaN(c_input)) knownCount++;
if (!isNaN(alpha_input)) knownCount++;
if (!isNaN(beta_input)) knownCount++;
var resultDiv = document.getElementById('triangleResult');
resultDiv.innerHTML = ";
if (knownCount < 2) {
resultDiv.innerHTML = 'Please enter at least two known values (sides or angles) to calculate the triangle.';
return;
}
var a = isNaN(a_input) ? null : a_input;
var b = isNaN(b_input) ? null : b_input;
var c = isNaN(c_input) ? null : c_input;
var alpha = isNaN(alpha_input) ? null : alpha_input; // in degrees
var beta = isNaN(beta_input) ? null : beta_input; // in degrees
var solved = false;
var maxIterations = 10; // To ensure all dependencies are resolved, but prevent infinite loops
for (var i = 0; i < maxIterations && !solved; i++) {
var initialA = a, initialB = b, initialC = c, initialAlpha = alpha, initialBeta = beta;
// — Angle Sum Property —
if (alpha !== null && beta === null) {
beta = 90 – alpha;
} else if (beta !== null && alpha === null) {
alpha = 90 – beta;
}
// — Pythagorean Theorem —
if (a !== null && b !== null && c === null) {
c = Math.sqrt(a * a + b * b);
} else if (a !== null && c !== null && b === null) {
if (c <= a) { resultDiv.innerHTML = 'Error: Hypotenuse (C) must be greater than Side A.'; return; }
b = Math.sqrt(c * c – a * a);
} else if (b !== null && c !== null && a === null) {
if (c 1 || a / c 1 || b / c 0, angles > 0 and < 90, sum of angles = 90)
if (a <= 0 || b <= 0 || c <= 0) {
resultDiv.innerHTML = 'Error: All sides must be positive values.';
return;
}
if (alpha = 90 || beta = 90) {
resultDiv.innerHTML = 'Error: Acute angles must be between 0 and 90 degrees.';
return;
}
if (Math.abs(alpha + beta – 90) > 0.01) { // Allow for floating point inaccuracies
resultDiv.innerHTML = 'Error: The sum of acute angles (Alpha + Beta) must be 90 degrees. Current sum: ' + (alpha + beta).toFixed(2) + ";
return;
}
// Also check Pythagorean theorem for consistency if all sides were given or derived
if (Math.abs(a*a + b*b – c*c) > 0.01) {
resultDiv.innerHTML = 'Error: Sides do not satisfy the Pythagorean theorem (a² + b² = c²). Please check your inputs.';
return;
}
resultDiv.innerHTML = '
Calculated Triangle Properties:
';
resultDiv.innerHTML += '
Side A: ' + a.toFixed(4) + ";
resultDiv.innerHTML += '
Side B: ' + b.toFixed(4) + ";
resultDiv.innerHTML += '
Hypotenuse C: ' + c.toFixed(4) + ";
resultDiv.innerHTML += '
Angle Alpha (opposite A): ' + alpha.toFixed(4) + ' degrees';
resultDiv.innerHTML += '
Angle Beta (opposite B): ' + beta.toFixed(4) + ' degrees';
resultDiv.innerHTML += '
Angle Gamma (right angle): 90 degrees';
}
function clearForm() {
document.getElementById('sideA').value = ";
document.getElementById('sideB').value = ";
document.getElementById('hypotenuseC').value = ";
document.getElementById('angleAlpha').value = ";
document.getElementById('angleBeta').value = ";
document.getElementById('triangleResult').innerHTML = ";
}