Triangle Angle Calculator

Triangle Angle Calculator: Unlocking the Geometry of Triangles

Understanding the angles within a triangle is fundamental to geometry, engineering, architecture, and many other fields. Whether you're a student learning trigonometry or a professional needing precise measurements, knowing how to calculate a triangle's angles is a crucial skill. Our Triangle Angle Calculator simplifies this process, allowing you to quickly determine the interior angles of any triangle given its three side lengths.

What is a Triangle?

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. The sum of the interior angles of any Euclidean triangle always equals 180 degrees.

How Does the Calculator Work? The Law of Cosines

This calculator primarily uses the Law of Cosines to determine the angles. The Law of Cosines is a fundamental theorem that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is a generalization of the Pythagorean theorem, which only applies to right-angled triangles.

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

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

By rearranging these formulas, we can solve for the angles:

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

Once you have the cosine of an angle, you can find the angle itself by taking the inverse cosine (arccos or cos⁻¹).

The Triangle Inequality Theorem

Before calculating angles, it's essential 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 valid triangle cannot be formed.

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

How to Use the Calculator

Simply enter the lengths of the three sides of your triangle into the respective fields. The calculator will then apply the Law of Cosines and the Triangle Inequality Theorem to determine if a valid triangle can be formed and, if so, calculate its interior angles in degrees.

Example Calculation

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

Side A = 5 units
Side B = 7 units
Side C = 9 units

First, check the Triangle Inequality Theorem:

5 + 7 > 9 (12 > 9, True)
5 + 9 > 7 (14 > 7, True)
7 + 9 > 5 (16 > 5, True)

Since all conditions are met, it's a valid triangle.

Now, calculate the angles using the Law of Cosines:

cos(A) = (7² + 9² - 5²) / (2 * 7 * 9) = (49 + 81 - 25) / 126 = 105 / 126 ≈ 0.8333
A = arccos(0.8333) ≈ 33.56 degrees
cos(B) = (5² + 9² - 7²) / (2 * 5 * 9) = (25 + 81 - 49) / 90 = 57 / 90 = 0.6333
B = arccos(0.6333) ≈ 50.70 degrees
cos(C) = (5² + 7² - 9²) / (2 * 5 * 7) = (25 + 49 - 81) / 70 = -7 / 70 = -0.1
C = arccos(-0.1) ≈ 95.74 degrees

Sum of angles: 33.56 + 50.70 + 95.74 = 180.00 degrees (approximately, due to rounding).

Side A Length:

Side B Length:

Side C Length:

Enter side lengths and click "Calculate Angles" to see the results.

function calculateTriangleAngles() { var a = parseFloat(document.getElementById('sideALength').value); var b = parseFloat(document.getElementById('sideBLength').value); var c = parseFloat(document.getElementById('sideCLength').value); var resultDiv = document.getElementById('triangleAngleResult'); // Input validation if (isNaN(a) || isNaN(b) || isNaN(c) || a <= 0 || b <= 0 || c c) && (a + c > b) && (b + c > a))) { resultDiv.innerHTML = 'The given side lengths do not form a valid triangle (Triangle Inequality Theorem violated).'; return; } // Calculate angles using Law of Cosines // 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 = (b * b + c * c – a * a) / (2 * b * c); var cosB = (a * a + c * c – b * b) / (2 * a * c); var cosC = (a * a + b * b – c * c) / (2 * a * b); // Handle potential floating point inaccuracies that might push cos values slightly out of [-1, 1] range 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); // 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; resultDiv.innerHTML = 'Calculated Angles:' + 'Angle A (opposite Side A): ' + angleA_deg.toFixed(2) + '°' + 'Angle B (opposite Side B): ' + angleB_deg.toFixed(2) + '°' + 'Angle C (opposite Side C): ' + angleC_deg.toFixed(2) + '°' + 'Sum of Angles: ' + sumOfAngles.toFixed(2) + '° (should be 180°)'; }