Calculate Angles of Triangle

Triangle Angle Calculator (SSS)

Side A Length: Side B Length: Side C Length:

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('triangleAnglesResult'); // Input validation if (isNaN(sideA) || isNaN(sideB) || isNaN(sideC) || sideA <= 0 || sideB <= 0 || sideC sideC) && (sideA + sideC > sideB) && (sideB + sideC > sideA))) { resultDiv.innerHTML = 'These side lengths do not form a valid triangle (Triangle Inequality Theorem).'; 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_val = (sideB * sideB + sideC * sideC – sideA * sideA) / (2 * sideB * sideC); var cosB_val = (sideA * sideA + sideC * sideC – sideB * sideB) / (2 * sideA * sideC); var cosC_val = (sideA * sideA + sideB * sideB – sideC * sideC) / (2 * sideA * sideB); // Clamp values to prevent Math.acos errors due to floating point inaccuracies cosA_val = Math.max(-1, Math.min(1, cosA_val)); cosB_val = Math.max(-1, Math.min(1, cosB_val)); cosC_val = Math.max(-1, Math.min(1, cosC_val)); var angleA_rad = Math.acos(cosA_val); var angleB_rad = Math.acos(cosB_val); var angleC_rad = Math.acos(cosC_val); // Convert radians to degrees 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; var output = '

Calculated Angles:

'; output += 'Angle A: ' + angleA_deg.toFixed(2) + '°'; output += 'Angle B: ' + angleB_deg.toFixed(2) + '°'; output += 'Angle C: ' + angleC_deg.toFixed(2) + '°'; output += '(Sum of angles: ' + sumOfAngles.toFixed(2) + '°)'; // Check if sum is approximately 180 (due to floating point precision) if (Math.abs(sumOfAngles – 180) > 0.01) { output += 'Warning: Sum of angles is not exactly 180°. This might be due to floating point precision or invalid input leading to a degenerate triangle.'; } resultDiv.innerHTML = output; } .calculator-container { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; font-family: Arial, sans-serif; } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 20px; } .calculator-inputs label { display: block; margin-bottom: 5px; font-weight: bold; color: #555; } .calculator-inputs input[type="number"] { width: calc(100% – 22px); padding: 10px; margin-bottom: 15px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calculator-inputs button { background-color: #007bff; color: white; padding: 12px 20px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; width: 100%; box-sizing: border-box; } .calculator-inputs button:hover { background-color: #0056b3; } .calculator-results { margin-top: 20px; padding: 15px; background-color: #e9ecef; border: 1px solid #dee2e6; border-radius: 4px; } .calculator-results h3 { color: #333; margin-top: 0; } .calculator-results p { margin-bottom: 5px; color: #333; }

Understanding Triangle Angles and How to Calculate Them

Triangles are fundamental geometric shapes, and understanding their properties, especially their angles, is crucial in various fields from architecture to engineering and even art. A triangle is a polygon with three edges and three vertices. A key property of any Euclidean triangle is that the sum of its interior angles always equals 180 degrees.

What is a Triangle?

At its core, a triangle is a three-sided polygon. It's the simplest polygon and forms the basis for understanding more complex shapes. Triangles can be classified by their side lengths (equilateral, isosceles, scalene) or by their angles (acute, right, obtuse).

Equilateral Triangle: All three sides are equal in length, and all three angles are equal (60° each).
Isosceles Triangle: Two sides are equal in length, and the angles opposite those sides are also equal.
Scalene Triangle: All three sides have different lengths, and all three angles are different.
Right Triangle: One angle measures exactly 90 degrees.
Acute Triangle: All three angles are acute (less than 90 degrees).
Obtuse Triangle: One angle is obtuse (greater than 90 degrees).

The Importance of Angles

The angles within a triangle dictate its shape and proportions. Knowing the angles allows us to solve for unknown side lengths, determine spatial relationships, and design structures. For instance, in surveying, triangulation is used to measure distances and angles between points.

How to Calculate Angles Given Three Side Lengths (SSS)

When you know the lengths of all three sides of a triangle (Side-Side-Side or SSS), you can calculate each of its interior angles using the Law of Cosines. This powerful trigonometric formula relates the lengths of the sides of a triangle to the cosine of one of its angles.

The Law of Cosines Formula:

For a triangle with sides a, b, and c, and angles A, B, and C opposite those respective sides:

To find Angle A: cos(A) = (b² + c² - a²) / (2bc)
To find Angle B: cos(B) = (a² + c² - b²) / (2ac)
To find Angle C: cos(C) = (a² + b² - c²) / (2ab)

Once you calculate the cosine value, you take the inverse cosine (arccos or cos⁻¹) to find the angle in radians, which then needs to be converted to degrees (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.

a + b > c
a + c > b
b + c > a

Using the Calculator

Our Triangle Angle Calculator simplifies this process. Simply input the lengths of the three sides (Side A, Side B, and Side C) into the respective fields. The calculator will then apply the Law of Cosines to determine the measure of each interior angle in degrees. It also performs a check using the Triangle Inequality Theorem to ensure the side lengths form a valid triangle.

Example Calculation:

Let's say you have a triangle with the following side lengths:

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

This is a classic example of a right-angled triangle (a 3-4-5 Pythagorean triple).

Using the Law of Cosines:

For Angle A:
cos(A) = (4² + 5² - 3²) / (2 * 4 * 5)
cos(A) = (16 + 25 - 9) / 40
cos(A) = 32 / 40 = 0.8
A = arccos(0.8) ≈ 36.87°
For Angle B:
cos(B) = (3² + 5² - 4²) / (2 * 3 * 5)
cos(B) = (9 + 25 - 16) / 30
cos(B) = 18 / 30 = 0.6
B = arccos(0.6) ≈ 53.13°
For Angle C:
cos(C) = (3² + 4² - 5²) / (2 * 3 * 4)
cos(C) = (9 + 16 - 25) / 24
cos(C) = 0 / 24 = 0
C = arccos(0) = 90.00°

The sum of the angles is approximately 36.87° + 53.13° + 90.00° = 180.00°, confirming a valid triangle.