Calculating the degree of an angle within a triangle is a fundamental concept in trigonometry. When you know the lengths of all three sides of a triangle (a situation known as SSS – Side-Side-Side), you can precisely determine any of its internal angles using the Law of Cosines.
The Law of Cosines
The Law of Cosines provides a relationship between the lengths of the sides of a triangle and the cosine of one of its angles. For a triangle with sides of length a, b, and c, and with angles A, B, and C opposite to those sides respectively, the law states:
c² = a² + b² - 2ab * cos(C)
b² = a² + c² - 2ac * cos(B)
a² = b² + c² - 2bc * cos(A)
To calculate an angle, we rearrange these formulas. For example, to find angle C:
Find the angle C using the inverse cosine function (arccos or cos⁻¹): C = arccos((a² + b² - c²) / (2ab))
This calculator uses these rearranged formulas to find the specified angle in degrees.
How This Calculator Works
This calculator takes the lengths of the three sides of a triangle (Side A, Side B, Side C) and allows you to select which angle you wish to calculate (the angle opposite Side A, Side B, or Side C). It then applies the Law of Cosines, rearranged to solve for the angle, and outputs the result in degrees.
When to Use This Calculator
Surveying: Determining precise boundaries or land features where direct angle measurement is difficult.
Navigation: Calculating bearings and courses.
Engineering and Architecture: Designing structures that require specific angles for stability or aesthetics.
Physics: Analyzing forces, vectors, and projectile motion.
Geometry and Mathematics Education: Helping students understand and apply trigonometric principles.
Example Calculation
Let's say you have a triangle with the following side lengths:
Side A = 7 units
Side B = 8 units
Side C = 10 units
If you want to calculate the angle opposite Side C (Angle C), the formula would be:
cos(C) = (7² + 8² - 10²) / (2 * 7 * 8)
cos(C) = (49 + 64 - 100) / (112)
cos(C) = (113 - 100) / 112
cos(C) = 13 / 112
cos(C) ≈ 0.11607
Using a calculator for the inverse cosine:
C = arccos(0.11607) ≈ 83.34 degrees
This calculator automates this process for you.
function calculateAngle() {
var sideA = parseFloat(document.getElementById("sideA").value);
var sideB = parseFloat(document.getElementById("sideB").value);
var sideC = parseFloat(document.getElementById("sideC").value);
var angleToFind = document.getElementById("angleToFind").value;
var resultDiv = document.getElementById("result");
// Clear previous results or error messages
resultDiv.innerHTML = 'Enter values to see the result';
resultDiv.style.backgroundColor = 'var(–success-green)'; // Reset color
// Validate inputs
if (isNaN(sideA) || isNaN(sideB) || isNaN(sideC) || sideA <= 0 || sideB <= 0 || sideC <= 0) {
resultDiv.innerHTML = 'Please enter valid positive numbers for all sides.';
resultDiv.style.backgroundColor = '#dc3545'; // Error color
return;
}
// Triangle Inequality Theorem check
if (sideA + sideB <= sideC || sideA + sideC <= sideB || sideB + sideC 1) cosAngle = 1;
if (cosAngle < -1) cosAngle = -1;
angleDegrees = Math.acos(cosAngle) * (180 / Math.PI);
resultDiv.innerHTML = angleDegrees.toFixed(2) + '°';
resultDiv.style.backgroundColor = 'var(–success-green)'; // Success color
} catch (e) {
resultDiv.innerHTML = 'Calculation error. Please check your inputs.';
resultDiv.style.backgroundColor = '#dc3545'; // Error color
}
}