Calculate Angles of a Triangle

Triangle Angle Calculator

function calculateTriangleAngles() { var sideA = parseFloat(document.getElementById('sideA').value); var sideB = parseFloat(document.getElementById('sideB').value); var sideC = parseFloat(document.getElementById('sideC').value); var resultDiv = document.getElementById('result'); var angleAOutput = document.getElementById('angleAOutput'); var angleBOutput = document.getElementById('angleBOutput'); var angleCOutput = document.getElementById('angleCOutput'); angleAOutput.textContent = '–'; angleBOutput.textContent = '–'; angleCOutput.textContent = '–'; if (isNaN(sideA) || isNaN(sideB) || isNaN(sideC) || sideA <= 0 || sideB <= 0 || sideC sideC) && (sideA + sideC > sideB) && (sideB + sideC > sideA))) { resultDiv.style.color = '#dc3545'; resultDiv.textContent = 'These side lengths do not form a valid triangle (Triangle Inequality Theorem violated).'; return; } // Law of Cosines to find angles // cos(A) = (b^2 + c^2 – a^2) / (2bc) // cos(B) = (a^2 + c^2 – b^2) / (2ac) // cos(C) = (a^2 + b^2 – c^2) / (2ab) var cosA = (sideB * sideB + sideC * sideC – sideA * sideA) / (2 * sideB * sideC); var cosB = (sideA * sideA + sideC * sideC – sideB * sideB) / (2 * sideA * sideC); var cosC = (sideA * sideA + sideB * sideB – sideC * sideC) / (2 * sideA * sideB); // Ensure values are within the valid range for acos (-1 to 1) due to floating point inaccuracies cosA = Math.max(-1, Math.min(1, cosA)); cosB = Math.max(-1, Math.min(1, cosB)); cosC = Math.max(-1, Math.min(1, cosC)); var angleA_rad = Math.acos(cosA); var angleB_rad = Math.acos(cosB); var angleC_rad = Math.acos(cosC); var angleA_deg = angleA_rad * (180 / Math.PI); var angleB_deg = angleB_rad * (180 / Math.PI); var angleC_deg = angleC_rad * (180 / Math.PI); var sumOfAngles = angleA_deg + angleB_deg + angleC_deg; angleAOutput.textContent = angleA_deg.toFixed(2); angleBOutput.textContent = angleB_deg.toFixed(2); angleCOutput.textContent = angleC_deg.toFixed(2); resultDiv.style.color = '#28a745'; resultDiv.textContent = 'Angles calculated successfully! Sum of angles: ' + sumOfAngles.toFixed(2) + ' degrees.'; }

Understanding Triangle Angles with the Law of Cosines

A triangle is one of the most fundamental shapes in geometry, defined by three straight sides and three angles. The sum of the interior angles of any triangle always equals 180 degrees. While you might know the angles, sometimes you only have the lengths of the sides, and you need to determine the angles. This is where the Law of Cosines becomes an invaluable tool.

What is the Law of Cosines?

The Law of Cosines is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It's a generalization of the Pythagorean theorem and is particularly useful when you know all three side lengths (SSS – Side-Side-Side) or two sides and the included angle (SAS – Side-Angle-Side).

For a triangle with sides a, b, and c, and angles A, B, and C opposite those sides respectively, the Law of Cosines can be stated as:

• a² = b² + c² - 2bc * cos(A)
• b² = a² + c² - 2ac * cos(B)
• c² = a² + b² - 2ab * cos(C)

Calculating Angles from Side Lengths

To find the angles when you know all three side lengths, we can rearrange the Law of Cosines formulas to solve for the cosine of each angle:

• cos(A) = (b² + c² - a²) / (2bc)
• cos(B) = (a² + c² - b²) / (2ac)
• cos(C) = (a² + b² - c²) / (2ab)

Once you have the cosine value for an angle, you can use the inverse cosine function (arccos or cos⁻¹) to find the angle itself. Most calculators and programming languages provide this function, often as Math.acos() in JavaScript, which returns the angle in radians. To convert radians to degrees, you multiply by 180/π.

The Triangle Inequality Theorem

Before attempting to calculate angles, it's crucial to ensure that the given side lengths can actually form a triangle. The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this condition is not met, a triangle cannot be formed with those side lengths.

• a + b > c
• a + c > b
• b + c > a

How to Use the Calculator

Our Triangle Angle Calculator simplifies this process for you:

1. Enter Side A Length: Input the length of the first side of your triangle.
2. Enter Side B Length: Input the length of the second side.
3. Enter Side C Length: Input the length of the third side.
4. Click "Calculate Angles": The calculator will apply the Law of Cosines and the Triangle Inequality Theorem.

The results will display the measure of Angle A, Angle B, and Angle C in degrees. It will also confirm if the side lengths form a valid triangle and show the sum of the calculated angles, which should be approximately 180 degrees (slight deviations may occur due to rounding).

Example Calculation:

Let's consider a right-angled triangle with sides 3, 4, and 5 (a classic Pythagorean triple).

• Side A = 3
• Side B = 4
• Side C = 5

Using the Law of Cosines:

• cos(A) = (4² + 5² - 3²) / (2 * 4 * 5) = (16 + 25 - 9) / 40 = 32 / 40 = 0.8
• A = arccos(0.8) ≈ 36.87 degrees
• cos(B) = (3² + 5² - 4²) / (2 * 3 * 5) = (9 + 25 - 16) / 30 = 18 / 30 = 0.6
• B = arccos(0.6) ≈ 53.13 degrees
• cos(C) = (3² + 4² - 5²) / (2 * 3 * 4) = (9 + 16 - 25) / 24 = 0 / 24 = 0
• C = arccos(0) = 90 degrees

The sum of angles is 36.87 + 53.13 + 90 = 180 degrees, confirming a valid triangle and accurate angle calculations.