Angle Opposite Side A
Angle Opposite Side B
Angle Opposite Side C
Calculated Angle:
Degrees
Understanding Angle Calculations in Triangles
Calculating angles within a triangle is a fundamental concept in geometry and trigonometry, with applications ranging from surveying and navigation to engineering and physics. When you know the lengths of all three sides of a triangle, you can precisely determine the measure of each interior angle. The primary tool for this is the Law of Cosines.
The Law of Cosines
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides of length a, b, and c, and angles A, B, and C opposite those sides respectively, the law states:
c² = a² + b² - 2ab * cos(C)
b² = a² + c² - 2ac * cos(B)
a² = b² + c² - 2bc * cos(A)
Deriving the Angle Formula
To calculate an angle when all sides are known, we can rearrange the Law of Cosines. For example, to find angle A, we rearrange the third equation:
a² = b² + c² - 2bc * cos(A) 2bc * cos(A) = b² + c² - a² cos(A) = (b² + c² - a²) / (2bc)
And finally, to find the angle A itself, we use the inverse cosine function (arccos or cos⁻¹):
A = arccos((b² + c² - a²) / (2bc))
Similarly, for angles B and C:
B = arccos((a² + c² - b²) / (2ac))
C = arccos((a² + b² - c²) / (2ab))
The result of the arccos function is typically in radians, which is then converted to degrees for easier interpretation (1 radian = 180/π degrees).
Computer Graphics: Rendering 3D environments and animations.
Physics: Analyzing projectile motion and forces.
Input Validation
For a valid triangle to exist, the sum of the lengths of any two sides must be greater than the length of the third side (Triangle Inequality Theorem). Additionally, all side lengths must be positive. This calculator includes checks to ensure valid inputs are provided before performing calculations.
function calculateAngle() {
var sideA = parseFloat(document.getElementById('sideA').value);
var sideB = parseFloat(document.getElementById('sideB').value);
var sideC = parseFloat(document.getElementById('sideC').value);
var angleType = document.getElementById('angleType').value;
var resultDiv = document.getElementById('result');
var resultUnitDiv = document.getElementById('resultUnit');
var resultContainer = document.getElementById('result-container');
var errorMessageDiv = document.getElementById('errorMessage');
errorMessageDiv.textContent = "; // Clear previous errors
resultContainer.style.display = 'none'; // Hide result container
// — Input Validation —
if (isNaN(sideA) || isNaN(sideB) || isNaN(sideC)) {
errorMessageDiv.textContent = 'Please enter valid numbers for all side lengths.';
return;
}
if (sideA <= 0 || sideB <= 0 || sideC sideC) && (sideA + sideC > sideB) && (sideB + sideC > sideA))) {
errorMessageDiv.textContent = 'The given side lengths do not form a valid triangle.';
return;
}
var cosValue;
var angleRadians;
var angleDegrees;
// — Calculation Logic —
if (angleType === 'A') {
// Calculate angle A using cos(A) = (b² + c² – a²) / (2bc)
cosValue = (sideB * sideB + sideC * sideC – sideA * sideA) / (2 * sideB * sideC);
} else if (angleType === 'B') {
// Calculate angle B using cos(B) = (a² + c² – b²) / (2ac)
cosValue = (sideA * sideA + sideC * sideC – sideB * sideB) / (2 * sideA * sideC);
} else { // angleType === 'C'
// Calculate angle C using cos(C) = (a² + b² – c²) / (2ab)
cosValue = (sideA * sideA + sideB * sideB – sideC * sideC) / (2 * sideA * sideB);
}
// Ensure cosValue is within the valid range [-1, 1] due to potential floating-point inaccuracies
if (cosValue > 1) {
cosValue = 1;
} else if (cosValue < -1) {
cosValue = -1;
}
// Calculate angle in radians using arccos
angleRadians = Math.acos(cosValue);
// Convert radians to degrees
angleDegrees = angleRadians * (180 / Math.PI);
// Display the result
resultDiv.textContent = angleDegrees.toFixed(2); // Display with 2 decimal places
resultUnitDiv.textContent = 'Degrees';
resultContainer.style.display = 'block';
}