Angle of a Triangle Calculator

Angle of a Triangle 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 angleAResult = document.getElementById('angleAResult'); var angleBResult = document.getElementById('angleBResult'); var angleCResult = document.getElementById('angleCResult'); var sumAnglesResult = document.getElementById('sumAnglesResult'); var errorResult = document.getElementById('errorResult'); angleAResult.innerHTML = 'Angle A: '; angleBResult.innerHTML = 'Angle B: '; angleCResult.innerHTML = 'Angle C: '; sumAnglesResult.innerHTML = "; errorResult.innerHTML = "; if (isNaN(sideA) || isNaN(sideB) || isNaN(sideC) || sideA <= 0 || sideB <= 0 || sideC sideC) && (sideA + sideC > sideB) && (sideB + sideC > sideA))) { errorResult.innerHTML = 'The given side lengths do not form a valid triangle.'; 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 [-1, 1] due to potential 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 sumAngles = angleA_deg + angleB_deg + angleC_deg; angleAResult.innerHTML = 'Angle A: ' + angleA_deg.toFixed(2) + '°'; angleBResult.innerHTML = 'Angle B: ' + angleB_deg.toFixed(2) + '°'; angleCResult.innerHTML = 'Angle C: ' + angleC_deg.toFixed(2) + '°'; sumAnglesResult.innerHTML = 'Sum of Angles: ' + sumAngles.toFixed(2) + '°'; }

Understanding Triangle Angles

A triangle is a fundamental polygon with three sides and three angles. The sum of the interior angles of any Euclidean triangle always equals 180 degrees. Knowing the angles of a triangle is crucial in various fields, from geometry and trigonometry to engineering and architecture.

How to Calculate Angles Given Side Lengths (SSS)

When you know the lengths of all three sides of a triangle (Side-Side-Side or SSS), you can determine all three interior angles using the Law of Cosines. This law provides a relationship between the sides of a triangle and the cosine of one of its angles.

The formulas derived from the Law of Cosines are:

• For Angle A: \( \cos(A) = \frac{b^2 + c^2 – a^2}{2bc} \)
• For Angle B: \( \cos(B) = \frac{a^2 + c^2 – b^2}{2ac} \)
• For Angle C: \( \cos(C) = \frac{a^2 + b^2 – c^2}{2ab} \)

Where 'a', 'b', and 'c' are the lengths of the sides opposite to angles A, B, and C, respectively. After calculating the cosine of each angle, you take the inverse cosine (arccos or \( \cos^{-1} \)) to find the angle in radians, which is then converted to degrees.

The Triangle Inequality Theorem

Before calculating angles, it's important 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.

• \( a + b > c \)
• \( a + c > b \)
• \( b + c > a \)

Using the Calculator

Our Angle of a Triangle 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 to ensure that the side lengths you've entered can indeed form a valid triangle.

Example Calculation:

Let's consider a triangle with the following side lengths:

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

Using the Law of Cosines:

• For Angle A: \( \cos(A) = \frac{4^2 + 5^2 – 3^2}{2 \times 4 \times 5} = \frac{16 + 25 – 9}{40} = \frac{32}{40} = 0.8 \)
• \( A = \cos^{-1}(0.8) \approx 36.87^\circ \)
• For Angle B: \( \cos(B) = \frac{3^2 + 5^2 – 4^2}{2 \times 3 \times 5} = \frac{9 + 25 – 16}{30} = \frac{18}{30} = 0.6 \)
• \( B = \cos^{-1}(0.6) \approx 53.13^\circ \)
• For Angle C: \( \cos(C) = \frac{3^2 + 4^2 – 5^2}{2 \times 3 \times 4} = \frac{9 + 16 – 25}{24} = \frac{0}{24} = 0 \)
• \( C = \cos^{-1}(0) = 90^\circ \)

The sum of these angles is \( 36.87^\circ + 53.13^\circ + 90^\circ = 180^\circ \), confirming a valid triangle. This specific example is a right-angled triangle, as one angle is exactly 90 degrees.