Triangle Angle Calculator
Calculated Angles:
Enter the lengths of the three sides of your triangle above and click "Calculate Angles" to see the results.
Calculated Angles:
Please correct the input errors to see the results.'; return; } // Triangle Inequality Theorem if (!((sideA + sideB > sideC) && (sideA + sideC > sideB) && (sideB + sideC > sideA))) { errorDiv.innerHTML = 'These side lengths do not form a valid triangle (Triangle Inequality Theorem violated).'; resultDiv.innerHTML = 'Calculated Angles:
The provided side lengths cannot form a triangle.'; return; } // Law of Cosines to find angles in radians // 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); // Due to floating point inaccuracies, cos values might slightly exceed [-1, 1]. Clamp them. 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); // Display results resultDiv.innerHTML = `Calculated Angles:
Angle A: ${angleA_deg.toFixed(2)} degrees Angle B: ${angleB_deg.toFixed(2)} degrees Angle C: ${angleC_deg.toFixed(2)} degrees (Sum of angles: ${(angleA_deg + angleB_deg + angleC_deg).toFixed(2)} degrees) `; }Understanding Triangle Angles and How to Calculate Them
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. Knowing the angles of a triangle is crucial in various fields, from architecture and engineering to navigation and computer graphics.
Why Calculate Triangle Angles?
Calculating triangle angles allows us to:
- Determine Triangle Type: Identify if a triangle is acute (all angles less than 90°), obtuse (one angle greater than 90°), or right-angled (one angle exactly 90°).
- Solve Geometric Problems: Essential for solving complex geometric constructions and proofs.
- Engineering and Design: Used in structural analysis, surveying, and designing components where precise angles are critical.
- Navigation: Triangulation techniques rely on angle calculations to pinpoint locations.
The Law of Cosines: Our Tool for Angle Calculation
When you know the lengths of all three sides of a triangle (Side-Side-Side or SSS), you can use the Law of Cosines to find each of its angles. The Law of Cosines is a generalization of the Pythagorean theorem and relates the lengths of the sides of a triangle to the cosine of one of its angles.
For a triangle with sides a, b, and c, and opposite angles A, B, and C respectively, the formulas are:
cos(A) = (b² + c² - a²) / (2bc)cos(B) = (a² + c² - b²) / (2ac)cos(C) = (a² + b² - c²) / (2ab)
Once you calculate the cosine of an angle, you take the inverse cosine (arccos) to find the angle itself, typically converted from radians to degrees for easier understanding.
The Triangle Inequality Theorem: A Crucial Check
Before calculating angles, it's vital 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, no triangle can be formed with those side lengths.
a + b > ca + c > bb + c > a
How to Use the Triangle Angle Calculator
Our calculator simplifies this process for you:
- Enter Side A Length: Input the length of the first side of your triangle.
- Enter Side B Length: Input the length of the second side.
- Enter Side C Length: Input the length of the third side.
- Click "Calculate Angles": The calculator will instantly apply the Law of Cosines and the Triangle Inequality Theorem.
The results will display Angle A (opposite Side A), Angle B (opposite Side B), and Angle C (opposite Side C) in degrees, along with a sum of the angles to help verify the calculation (it should be very close to 180 degrees).
Examples of Triangle Angle Calculations
Example 1: A Right-Angled Triangle
Let's consider a classic right-angled triangle with sides 3, 4, and 5.
- Side A = 3
- Side B = 4
- Side C = 5
Using the calculator, you would find the angles to be approximately:
- Angle A ≈ 36.87 degrees
- Angle B ≈ 53.13 degrees
- Angle C ≈ 90.00 degrees
The sum is 36.87 + 53.13 + 90.00 = 180.00 degrees.
Example 2: An Equilateral Triangle
An equilateral triangle has all sides equal, and all angles equal to 60 degrees.
- Side A = 5
- Side B = 5
- Side C = 5
The calculator will show:
- Angle A = 60.00 degrees
- Angle B = 60.00 degrees
- Angle C = 60.00 degrees
The sum is 60.00 + 60.00 + 60.00 = 180.00 degrees.
Example 3: An Isosceles Triangle
Consider an isosceles triangle with two equal sides and a different third side.
- Side A = 5
- Side B = 5
- Side C = 8
The calculated angles would be:
- Angle A ≈ 38.68 degrees
- Angle B ≈ 38.68 degrees
- Angle C ≈ 102.64 degrees
The sum is 38.68 + 38.68 + 102.64 = 180.00 degrees.
This calculator provides a quick and accurate way to determine the angles of any valid triangle, making complex geometric calculations straightforward and accessible.