Use this calculator to find the missing angles and side lengths of a right-angled triangle. Enter any two known side lengths, and the calculator will determine the third side and both acute angles (in degrees).
Results:
Calculated Side A:
Calculated Side B:
Calculated Hypotenuse:
Angle Alpha (opposite Side A): degrees
Angle Beta (opposite Side B): degrees
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display: block;
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.calc-input-group input[type="number"] {
width: calc(100% – 22px);
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function calculateTriangleAngles() {
var sideA_str = document.getElementById("inputSideA").value;
var sideB_str = document.getElementById("inputSideB").value;
var hypotenuse_str = document.getElementById("inputHypotenuse").value;
var a = parseFloat(sideA_str);
var b = parseFloat(sideB_str);
var h = parseFloat(hypotenuse_str);
var countValidInputs = 0;
if (!isNaN(a) && a > 0) countValidInputs++;
if (!isNaN(b) && b > 0) countValidInputs++;
if (!isNaN(h) && h > 0) countValidInputs++;
var errorMessage = document.getElementById("errorMessage");
var outputSideA = document.getElementById("outputSideA");
var outputSideB = document.getElementById("outputSideB");
var outputHypotenuse = document.getElementById("outputHypotenuse");
var outputAngleAlpha = document.getElementById("outputAngleAlpha");
var outputAngleBeta = document.getElementById("outputAngleBeta");
errorMessage.textContent = "";
outputSideA.textContent = "";
outputSideB.textContent = "";
outputHypotenuse.textContent = "";
outputAngleAlpha.textContent = "";
outputAngleBeta.textContent = "";
if (countValidInputs 0 && !isNaN(b) && b > 0) {
calculatedA = a;
calculatedB = b;
calculatedH = Math.sqrt(a * a + b * b);
alphaRad = Math.atan(a / b);
betaRad = Math.atan(b / a);
}
// Case 2: Side A and Hypotenuse are given
else if (!isNaN(a) && a > 0 && !isNaN(h) && h > 0) {
if (a >= h) {
errorMessage.textContent = "Side A must be shorter than the Hypotenuse.";
return;
}
calculatedA = a;
calculatedH = h;
calculatedB = Math.sqrt(h * h – a * a);
alphaRad = Math.asin(a / h);
betaRad = Math.acos(a / h);
}
// Case 3: Side B and Hypotenuse are given
else if (!isNaN(b) && b > 0 && !isNaN(h) && h > 0) {
if (b >= h) {
errorMessage.textContent = "Side B must be shorter than the Hypotenuse.";
return;
}
calculatedB = b;
calculatedH = h;
calculatedA = Math.sqrt(h * h – b * b);
alphaRad = Math.acos(b / h);
betaRad = Math.asin(b / h);
}
else {
errorMessage.textContent = "Please enter two valid positive side lengths.";
return;
}
var alphaDegrees = alphaRad * (180 / Math.PI);
var betaDegrees = betaRad * (180 / Math.PI);
outputSideA.textContent = calculatedA.toFixed(3);
outputSideB.textContent = calculatedB.toFixed(3);
outputHypotenuse.textContent = calculatedH.toFixed(3);
outputAngleAlpha.textContent = alphaDegrees.toFixed(3);
outputAngleBeta.textContent = betaDegrees.toFixed(3);
}
function clearTriangleInputs() {
document.getElementById("inputSideA").value = "";
document.getElementById("inputSideB").value = "";
document.getElementById("inputHypotenuse").value = "";
document.getElementById("errorMessage").textContent = "";
document.getElementById("outputSideA").textContent = "";
document.getElementById("outputSideB").textContent = "";
document.getElementById("outputHypotenuse").textContent = "";
document.getElementById("outputAngleAlpha").textContent = "";
document.getElementById("outputAngleBeta").textContent = "";
}
Understanding Right-Angled Triangles and Their Angles
A right-angled triangle is a fundamental shape in geometry, characterized by one angle measuring exactly 90 degrees (a right angle). The sides of a right-angled triangle have specific names and relationships that are crucial for many calculations in mathematics, physics, engineering, and architecture.
Key Components of a Right-Angled Triangle:
Hypotenuse: This is the longest side of the triangle and is always opposite the right angle.
Opposite Side: For a given acute angle, the opposite side is the one directly across from it.
Adjacent Side: For a given acute angle, the adjacent side is the one next to it, which is not the hypotenuse.
The Pythagorean Theorem:
One of the most famous theorems related to right-angled triangles is the Pythagorean Theorem, which states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Mathematically, this is expressed as:
a² + b² = c²
This theorem allows us to find the length of any side if the other two sides are known.
Trigonometric Ratios (SOH CAH TOA):
To find the angles of a right-angled triangle, we use trigonometric ratios, often remembered by the acronym SOH CAH TOA:
SOH:Sine = Opposite / Hypotenuse
CAH:Cosine = Adjacent / Hypotenuse
TOA:Tangent = Opposite / Adjacent
By using the inverse trigonometric functions (arcsin, arccos, arctan), we can calculate the angles when the side lengths are known.
How to Use the Right-Angled Triangle Angle Calculator:
Our calculator simplifies the process of finding missing angles and sides. Here's how to use it:
Identify Known Sides: Look at your right-angled triangle and determine which two side lengths you know.
Enter Values: Input the lengths of the two known sides into the corresponding fields (Side A, Side B, or Hypotenuse). Leave the unknown side field blank.
Calculate: Click the "Calculate Angles" button.
View Results: The calculator will display the length of the third side and the measures of Angle Alpha (opposite Side A) and Angle Beta (opposite Side B) in degrees.
Example Calculation:
Let's say you have a right-angled triangle where:
Side A (Opposite Angle Alpha) = 3 units
Side B (Opposite Angle Beta) = 4 units
Using the calculator:
Enter '3' in the "Side A" field.
Enter '4' in the "Side B" field.
Click "Calculate Angles".
The calculator will output:
Calculated Hypotenuse: 5.000 units (from 3² + 4² = 5²)
Angle Alpha: 36.870 degrees (from arctan(3/4))
Angle Beta: 53.130 degrees (from arctan(4/3))
Notice that Angle Alpha + Angle Beta + 90 degrees (the right angle) = 36.870 + 53.130 + 90 = 180 degrees, confirming the sum of angles in a triangle.
Applications:
Understanding and calculating angles in right-angled triangles is essential in various fields:
Construction and Architecture: For designing structures, calculating roof pitches, and ensuring stability.
Navigation: Used in determining distances, bearings, and positions.
Physics: Analyzing forces, trajectories, and vectors.
Engineering: In mechanical design, electrical circuits, and civil engineering projects.
Surveying: Measuring land, elevations, and distances.
This calculator provides a quick and accurate way to perform these fundamental trigonometric calculations, making complex problems more accessible.