Understanding Right Angled Triangles and the Calculator
A right-angled triangle, also known as a right triangle, is a fundamental shape in geometry. It's defined by having one interior angle that measures exactly 90 degrees (a right angle). The sides opposite the acute angles are called the 'legs' (often referred to as side A and side B, or adjacent and opposite relative to an angle), and the side opposite the right angle is the longest side, called the 'hypotenuse'.
Key Properties and Formulas:
Pythagorean Theorem: This is the cornerstone for calculations involving the sides of a right-angled triangle. It states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).
Formula: a² + b² = c² This allows us to find any side if the other two are known.
Trigonometric Ratios (SOH CAH TOA): These ratios relate the angles of a right-angled triangle to the lengths of its sides.
Sine (sin):sin(angle) = Opposite / Hypotenuse
Cosine (cos):cos(angle) = Adjacent / Hypotenuse
Tangent (tan):tan(angle) = Opposite / Adjacent
These are crucial for finding angles when sides are known, or sides when an angle and a side are known.
Sum of Angles: The sum of the interior angles in any triangle is always 180 degrees. In a right-angled triangle, one angle is 90 degrees, so the other two acute angles must add up to 90 degrees.
Formula: Angle A + Angle B + 90° = 180°, which simplifies to Angle A + Angle B = 90°.
How the Calculator Works:
This calculator is designed to find missing sides and angles of a right-angled triangle. You can input known values, and it will calculate the rest. The calculator utilizes the Pythagorean theorem and trigonometric functions (sine, cosine, tangent) to perform its calculations. It assumes inputs are in consistent units for length and degrees for angles.
Use Cases:
Construction & Architecture: Ensuring square corners, calculating roof slopes, and determining diagonal lengths for framing.
Navigation: Calculating distances and bearings.
Engineering: Analyzing forces and structures.
Physics: Solving problems involving vectors and projectile motion.
Mathematics Education: Helping students visualize and solve geometry problems.
Remember to ensure your inputs are accurate. If you input side lengths, the calculator will determine the hypotenuse and the two acute angles. If you input one side length and one acute angle, it will calculate the other side length, the hypotenuse, and the remaining angle.
function getInputValue(id) {
var value = parseFloat(document.getElementById(id).value);
return isNaN(value) ? null : value;
}
function formatResult(value, label) {
if (value === null || isNaN(value)) {
return '—' + label + '';
}
// Basic rounding for display
var roundedValue = Math.round(value * 100) / 100;
return '' + roundedValue.toLocaleString() + '' + label + '';
}
function resetCalculator() {
document.getElementById('sideA').value = ";
document.getElementById('sideB').value = ";
document.getElementById('angleA').value = ";
document.getElementById('angleB').value = ";
document.getElementById('resultsOutput').innerHTML = formatResult(null, 'Hypotenuse (C)');
document.getElementById('resultsOutputAngleA').innerHTML = formatResult(null, 'Angle A (degrees)');
document.getElementById('resultsOutputAngleB').innerHTML = formatResult(null, 'Angle B (degrees)');
document.getElementById('resultsOutputSideA').innerHTML = formatResult(null, 'Side A (Adjacent)');
document.getElementById('resultsOutputSideB').innerHTML = formatResult(null, 'Side B (Opposite)');
}
// — Calculation Functions —
function calculateHypotenuseAndAngles() {
var sideA = getInputValue('sideA');
var sideB = getInputValue('sideB');
var angleA = getInputValue('angleA'); // Assuming angle A is opposite side B
var angleB = getInputValue('angleB'); // Assuming angle B is opposite side A
var results = {};
// If sideA and sideB are provided
if (sideA !== null && sideB !== null) {
results.hypotenuse = Math.sqrt(Math.pow(sideA, 2) + Math.pow(sideB, 2));
results.angleA = (Math.atan(sideB / sideA) * 180 / Math.PI);
results.angleB = 90 – results.angleA;
}
// If sideA and angleA are provided (angleA opposite sideB)
else if (sideA !== null && angleA !== null && angleA 0) {
results.sideB = sideA * Math.tan(angleA * Math.PI / 180);
results.hypotenuse = sideA / Math.cos(angleA * Math.PI / 180);
results.angleB = 90 – angleA;
}
// If sideB and angleA are provided (angleA opposite sideB)
else if (sideB !== null && angleA !== null && angleA 0) {
results.sideA = sideB / Math.tan(angleA * Math.PI / 180);
results.hypotenuse = sideB / Math.sin(angleA * Math.PI / 180);
results.angleB = 90 – angleA;
}
// If sideA and angleB are provided (angleB opposite sideA)
else if (sideA !== null && angleB !== null && angleB 0) {
results.sideB = sideA * Math.tan(angleB * Math.PI / 180);
results.hypotenuse = sideA / Math.cos(angleB * Math.PI / 180);
results.angleA = 90 – angleB;
}
// If sideB and angleB are provided (angleB opposite sideA)
else if (sideB !== null && angleB !== null && angleB 0) {
results.sideA = sideB / Math.tan(angleB * Math.PI / 180);
results.hypotenuse = sideB / Math.sin(angleB * Math.PI / 180);
results.angleA = 90 – angleB;
}
// If hypotenuse and sideA are provided
else if (results.hypotenuse === undefined && sideA !== null) {
var hypotenuse = getInputValue('hypotenuse'); // Assuming hypotenuse input exists or is calculated
if (hypotenuse !== null && hypotenuse > sideA) {
results.hypotenuse = hypotenuse;
results.sideB = Math.sqrt(Math.pow(hypotenuse, 2) – Math.pow(sideA, 2));
results.angleA = (Math.asin(sideA / hypotenuse) * 180 / Math.PI); // Angle A opposite side A? No, angle A opposite side B. sin(A)=B/C, cos(A)=A/C
results.angleA = (Math.acos(sideA / hypotenuse) * 180 / Math.PI); // Angle A adjacent to side A
results.angleB = 90 – results.angleA;
}
}
// If hypotenuse and sideB are provided
else if (results.hypotenuse === undefined && sideB !== null) {
var hypotenuse = getInputValue('hypotenuse');
if (hypotenuse !== null && hypotenuse > sideB) {
results.hypotenuse = hypotenuse;
results.sideA = Math.sqrt(Math.pow(hypotenuse, 2) – Math.pow(sideB, 2));
results.angleB = (Math.acos(sideB / hypotenuse) * 180 / Math.PI); // Angle B adjacent to side B
results.angleA = 90 – results.angleB;
}
}
// If hypotenuse and angleA are provided
else if (results.hypotenuse === undefined && angleA !== null && angleA 0) {
var hypotenuse = getInputValue('hypotenuse');
if (hypotenuse !== null) {
results.hypotenuse = hypotenuse;
results.sideA = hypotenuse * Math.cos(angleA * Math.PI / 180);
results.sideB = hypotenuse * Math.sin(angleA * Math.PI / 180);
results.angleB = 90 – angleA;
}
}
// If hypotenuse and angleB are provided
else if (results.hypotenuse === undefined && angleB !== null && angleB 0) {
var hypotenuse = getInputValue('hypotenuse');
if (hypotenuse !== null) {
results.hypotenuse = hypotenuse;
results.sideB = hypotenuse * Math.cos(angleB * Math.PI / 180);
results.sideA = hypotenuse * Math.sin(angleB * Math.PI / 180);
results.angleA = 90 – angleB;
}
}
document.getElementById('resultsOutput').innerHTML = formatResult(results.hypotenuse, 'Hypotenuse (C)');
document.getElementById('resultsOutputAngleA').innerHTML = formatResult(results.angleA, 'Angle A (degrees)');
document.getElementById('resultsOutputAngleB').innerHTML = formatResult(results.angleB, 'Angle B (degrees)');
document.getElementById('resultsOutputSideA').innerHTML = formatResult(results.sideA, 'Side A (Adjacent)');
document.getElementById('resultsOutputSideB').innerHTML = formatResult(results.sideB, 'Side B (Opposite)');
}
function calculateAngleBAndHypotenuse() {
var sideA = getInputValue('sideA');
var angleA = getInputValue('angleA'); // Angle A opposite side B
var results = {};
if (sideA !== null && angleA !== null && angleA > 0 && angleA 0 && angleB < 90) {
results.angleA = 90 – angleB;
results.hypotenuse = sideB / Math.cos(angleB * Math.PI / 180); // cos(B) = sideB / hypotenuse
results.sideA = sideB * Math.tan(angleB * Math.PI / 180); // tan(B) = sideA / sideB
} else if (sideB !== null && getInputValue('sideA') !== null) {
var sideA = getInputValue('sideA');
results.hypotenuse = Math.sqrt(Math.pow(sideA, 2) + Math.pow(sideB, 2));
results.angleB = (Math.atan(sideA / sideB) * 180 / Math.PI);
results.angleA = 90 – results.angleB;
}
document.getElementById('resultsOutput').innerHTML = formatResult(results.hypotenuse, 'Hypotenuse (C)');
document.getElementById('resultsOutputAngleA').innerHTML = formatResult(results.angleA, 'Angle A (degrees)');
document.getElementById('resultsOutputAngleB').innerHTML = formatResult(results.angleB, 'Angle B (degrees)');
document.getElementById('resultsOutputSideA').innerHTML = formatResult(results.sideA, 'Side A (Adjacent)');
document.getElementById('resultsOutputSideB').innerHTML = formatResult(sideB, 'Side B (Opposite)'); // Keep original sideB
}