Angle of Right Angled Triangle Calculator

Right-Angled Triangle Angle Calculator

Enter exactly two positive side lengths to calculate the angles and the remaining side of a right-angled triangle.

function calculateAngles() { var sideA_val = parseFloat(document.getElementById('sideA').value); var sideB_val = parseFloat(document.getElementById('sideB').value); var hypotenuseC_val = parseFloat(document.getElementById('hypotenuseC').value); var inputs = []; if (!isNaN(sideA_val) && sideA_val > 0) inputs.push({ id: 'sideA', value: sideA_val }); if (!isNaN(sideB_val) && sideB_val > 0) inputs.push({ id: 'sideB', value: sideB_val }); if (!isNaN(hypotenuseC_val) && hypotenuseC_val > 0) inputs.push({ id: 'hypotenuseC', value: hypotenuseC_val }); var resultDiv = document.getElementById('result'); resultDiv.innerHTML = "; // Clear previous results if (inputs.length !== 2) { resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.borderColor = '#f5c6cb'; resultDiv.style.color = '#721c24'; resultDiv.innerHTML = 'Error: Please enter exactly two positive side lengths to calculate the triangle\'s angles and remaining side.'; return; } var angleA, angleB, calcSideA, calcSideB, calcHypotenuseC; var PI = Math.PI; // Determine which two sides were provided var input1 = inputs[0]; var input2 = inputs[1]; // Case 1: Side A and Side B are given if ((input1.id === 'sideA' && input2.id === 'sideB') || (input1.id === 'sideB' && input2.id === 'sideA')) { var a = sideA_val; var b = sideB_val; angleA = Math.atan(a / b) * (180 / PI); angleB = 90 – angleA; calcHypotenuseC = Math.sqrt(a * a + b * b); calcSideA = a; calcSideB = b; } // Case 2: Side A and Hypotenuse C are given else if ((input1.id === 'sideA' && input2.id === 'hypotenuseC') || (input1.id === 'hypotenuseC' && input2.id === 'sideA')) { var a = sideA_val; var c = hypotenuseC_val; if (a >= c) { resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.borderColor = '#f5c6cb'; resultDiv.style.color = '#721c24'; resultDiv.innerHTML = 'Error: Side A cannot be greater than or equal to the Hypotenuse.'; return; } angleA = Math.asin(a / c) * (180 / PI); angleB = 90 – angleA; calcSideB = Math.sqrt(c * c – a * a); calcSideA = a; calcHypotenuseC = c; } // Case 3: Side B and Hypotenuse C are given else if ((input1.id === 'sideB' && input2.id === 'hypotenuseC') || (input1.id === 'hypotenuseC' && input2.id === 'sideB')) { var b = sideB_val; var c = hypotenuseC_val; if (b >= c) { resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.borderColor = '#f5c6cb'; resultDiv.style.color = '#721c24'; resultDiv.innerHTML = 'Error: Side B cannot be greater than or equal to the Hypotenuse.'; return; } angleB = Math.asin(b / c) * (180 / PI); // Angle B is opposite side B angleA = 90 – angleB; calcSideA = Math.sqrt(c * c – b * b); calcSideB = b; calcHypotenuseC = c; } else { resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.borderColor = '#f5c6cb'; resultDiv.style.color = '#721c24'; resultDiv.innerHTML = 'Error: An unexpected error occurred. Please check your inputs.'; return; } resultDiv.style.backgroundColor = '#e9f7ef'; resultDiv.style.borderColor = '#d4edda'; resultDiv.style.color = '#155724'; resultDiv.innerHTML = '

Calculation Results:

' + 'Angle A (opposite Side A): ' + angleA.toFixed(2) + '°' + 'Angle B (opposite Side B): ' + angleB.toFixed(2) + '°' + 'Angle C (Right Angle): 90.00°' + 'Side A: ' + calcSideA.toFixed(2) + " + 'Side B: ' + calcSideB.toFixed(2) + " + 'Hypotenuse (Side C): ' + calcHypotenuseC.toFixed(2) + "; }

Understanding the Right-Angled Triangle Angle Calculator

A right-angled triangle is a fundamental shape in geometry, characterized by one angle measuring exactly 90 degrees. The sides adjacent to the right angle are called legs, and the side opposite the right angle is known as the hypotenuse, which is always the longest side. This calculator helps you determine the unknown angles and the remaining side of a right-angled triangle when you provide any two of its side lengths.

Key Concepts for Right-Angled Triangles

To understand how this calculator works, it's essential to grasp a few core mathematical principles:

1. Pythagorean Theorem: This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this is expressed as a² + b² = c². This allows us to find an unknown side if two sides are known.
2. Sum of Angles: The sum of all interior angles in any triangle is always 180 degrees. Since a right-angled triangle already has one 90-degree angle, the sum of the other two acute angles must be 90 degrees (e.g., Angle A + Angle B = 90°).
3. Trigonometric Ratios (SOH CAH TOA): These ratios relate the angles of a right-angled triangle to the ratios of its side lengths.
   • Sine (Sin): Opposite / Hypotenuse (SOH)
   • Cosine (Cos): Adjacent / Hypotenuse (CAH)
   • Tangent (Tan): Opposite / Adjacent (TOA)
   The inverse trigonometric functions (arcsin, arccos, arctan) are used to find the angles when the side ratios are known.

How the Calculator Works

This calculator uses the principles above to solve for the unknown angles and side. Here's a breakdown of the scenarios it handles:

• If you provide Side A (Opposite) and Side B (Adjacent):

  The calculator first finds Angle A using the tangent function: Angle A = arctan(Side A / Side B). Then, Angle B is found by subtracting Angle A from 90 degrees: Angle B = 90° - Angle A. Finally, the Hypotenuse (Side C) is calculated using the Pythagorean theorem: Hypotenuse = √(Side A² + Side B²).

  Example: If Side A = 3 and Side B = 4, then Angle A = arctan(3/4) ≈ 36.87°, Angle B = 90° – 36.87° = 53.13°, and Hypotenuse = √(3² + 4²) = √25 = 5.

• If you provide Side A (Opposite) and Hypotenuse (Side C):

  Angle A is calculated using the sine function: Angle A = arcsin(Side A / Hypotenuse). Angle B is then 90° - Angle A. Side B (Adjacent) is found using the Pythagorean theorem: Side B = √(Hypotenuse² - Side A²).

  Example: If Side A = 6 and Hypotenuse = 10, then Angle A = arcsin(6/10) ≈ 36.87°, Angle B = 90° – 36.87° = 53.13°, and Side B = √(10² – 6²) = √(100 – 36) = √64 = 8.

• If you provide Side B (Opposite Angle B) and Hypotenuse (Side C):

  Angle B is calculated using the sine function: Angle B = arcsin(Side B / Hypotenuse). Angle A is then 90° - Angle B. Side A (Opposite Angle A) is found using the Pythagorean theorem: Side A = √(Hypotenuse² - Side B²).

  Example: If Side B = 8 and Hypotenuse = 10, then Angle B = arcsin(8/10) ≈ 53.13°, Angle A = 90° – 53.13° = 36.87°, and Side A = √(10² – 8²) = √(100 – 64) = √36 = 6.

The calculator ensures that your inputs are valid (positive numbers, and for hypotenuse calculations, that the leg is not longer than or equal to the hypotenuse) to prevent impossible triangle scenarios.

Why Use This Calculator?

This tool is invaluable for students, engineers, architects, carpenters, and anyone working with geometric problems. It simplifies complex trigonometric calculations, saving time and reducing the potential for errors. Whether you're designing a roof, calculating distances, or solving a geometry homework problem, this calculator provides quick and accurate results for right-angled triangles.