Angles of a Right Triangle Calculator

Right Triangle Angles Calculator

Results:

Enter at least two side lengths above to calculate the angles of your right triangle.

function calculateAngles() { var sideA = parseFloat(document.getElementById('sideALength').value); var sideB = parseFloat(document.getElementById('sideBLength').value); var hypotenuse = parseFloat(document.getElementById('hypotenuseLength').value); var resultDiv = document.getElementById('angleResults'); resultDiv.innerHTML = '

Results:

'; // Clear previous results var inputsProvided = 0; if (!isNaN(sideA) && sideA > 0) inputsProvided++; if (!isNaN(sideB) && sideB > 0) inputsProvided++; if (!isNaN(hypotenuse) && hypotenuse > 0) inputsProvided++; if (inputsProvided 0 && !isNaN(sideB) && sideB > 0) { angleA_rad = Math.atan(sideA / sideB); angleA_deg = angleA_rad * (180 / Math.PI); angleB_deg = 90 – angleA_deg; calculatedHypotenuse = Math.sqrt(sideA * sideA + sideB * sideB); if (!isNaN(hypotenuse) && hypotenuse > 0 && Math.abs(calculatedHypotenuse – hypotenuse) > 0.01) { // Tolerance for floating point resultDiv.innerHTML += 'Warning: Provided side lengths (A, B, Hypotenuse) are inconsistent with Pythagorean theorem. Calculations are based on Side A and Side B.'; } resultDiv.innerHTML += 'Angle A: ' + angleA_deg.toFixed(2) + '°'; resultDiv.innerHTML += 'Angle B: ' + angleB_deg.toFixed(2) + '°'; resultDiv.innerHTML += 'Angle C (Right Angle): 90.00°'; resultDiv.innerHTML += 'Hypotenuse (calculated): ' + calculatedHypotenuse.toFixed(2) + "; return; } // Case 2: Side A and Hypotenuse are provided if (!isNaN(sideA) && sideA > 0 && !isNaN(hypotenuse) && hypotenuse > 0) { if (sideA >= hypotenuse) { resultDiv.innerHTML += 'Error: Side A cannot be greater than or equal to the Hypotenuse.'; return; } angleA_rad = Math.asin(sideA / hypotenuse); angleA_deg = angleA_rad * (180 / Math.PI); angleB_deg = 90 – angleA_deg; calculatedSideB = Math.sqrt(hypotenuse * hypotenuse – sideA * sideA); if (!isNaN(sideB) && sideB > 0 && Math.abs(calculatedSideB – sideB) > 0.01) { // Tolerance for floating point resultDiv.innerHTML += 'Warning: Provided side lengths (A, B, Hypotenuse) are inconsistent with Pythagorean theorem. Calculations are based on Side A and Hypotenuse.'; } resultDiv.innerHTML += 'Angle A: ' + angleA_deg.toFixed(2) + '°'; resultDiv.innerHTML += 'Angle B: ' + angleB_deg.toFixed(2) + '°'; resultDiv.innerHTML += 'Angle C (Right Angle): 90.00°'; resultDiv.innerHTML += 'Side B (calculated): ' + calculatedSideB.toFixed(2) + "; return; } // Case 3: Side B and Hypotenuse are provided if (!isNaN(sideB) && sideB > 0 && !isNaN(hypotenuse) && hypotenuse > 0) { if (sideB >= hypotenuse) { resultDiv.innerHTML += 'Error: Side B cannot be greater than or equal to the Hypotenuse.'; return; } angleA_rad = Math.acos(sideB / hypotenuse); // cos(A) = adjacent/hypotenuse = sideB/hypotenuse angleA_deg = angleA_rad * (180 / Math.PI); angleB_deg = 90 – angleA_deg; calculatedSideA = Math.sqrt(hypotenuse * hypotenuse – sideB * sideB); if (!isNaN(sideA) && sideA > 0 && Math.abs(calculatedSideA – sideA) > 0.01) { // Tolerance for floating point resultDiv.innerHTML += 'Warning: Provided side lengths (A, B, Hypotenuse) are inconsistent with Pythagorean theorem. Calculations are based on Side B and Hypotenuse.'; } resultDiv.innerHTML += 'Angle A: ' + angleA_deg.toFixed(2) + '°'; resultDiv.innerHTML += 'Angle B: ' + angleB_deg.toFixed(2) + '°'; resultDiv.innerHTML += 'Angle C (Right Angle): 90.00°'; resultDiv.innerHTML += 'Side A (calculated): ' + calculatedSideA.toFixed(2) + "; return; } // Fallback if only one or zero valid inputs were provided (should be caught by inputsProvided check) resultDiv.innerHTML += 'Please enter at least two positive side lengths to calculate the angles.'; }

Understanding the Angles of a Right Triangle

A right triangle is a special type of triangle that contains one angle exactly equal to 90 degrees (a right angle). This unique property makes it fundamental in geometry, trigonometry, and various real-world applications, from construction to navigation.

Key Components of a Right Triangle

• Right Angle (90°): Always present in a right triangle. We typically label this Angle C.
• Hypotenuse: The longest side of the right triangle, always opposite the right angle.
• Opposite Side: The side directly across from a given acute angle.
• Adjacent Side: The side next to a given acute angle that is not the hypotenuse.
• Acute Angles: The other two angles in a right triangle are always acute (less than 90 degrees). The sum of these two acute angles is always 90 degrees. If we label them Angle A and Angle B, then A + B = 90°.

Trigonometric Ratios: SOH CAH TOA

To find the angles of a right triangle when you know the lengths of its sides, we use trigonometric ratios. These ratios relate the angles of a right triangle to the lengths of its sides:

• Sine (SOH): sin(Angle) = Opposite / Hypotenuse
• Cosine (CAH): cos(Angle) = Adjacent / Hypotenuse
• Tangent (TOA): tan(Angle) = Opposite / Adjacent

To find the angle itself, we use the inverse trigonometric functions: arcsin (or sin⁻¹), arccos (or cos⁻¹), and arctan (or tan⁻¹).

How to Use the Calculator

Our Right Triangle Angles Calculator simplifies the process of finding the unknown acute angles. You only need to provide the lengths of any two sides of the right triangle. The calculator will automatically determine which sides you've entered and apply the correct trigonometric formula.

Here's what each input represents:

• Side A Length (Opposite Angle A): The length of the side opposite the angle you're calling 'Angle A'.
• Side B Length (Adjacent to Angle A, Opposite Angle B): The length of the side adjacent to 'Angle A' (and opposite 'Angle B').
• Hypotenuse Length: The length of the longest side, opposite the 90-degree angle.

Simply enter positive numerical values for any two of these fields, and click "Calculate Angles". The calculator will then display the measures of Angle A, Angle B, and confirm the 90-degree Angle C.

Examples of Angle Calculation

Example 1: Given Opposite and Adjacent Sides

Let's say you have a right triangle where:

• Side A (Opposite Angle A) = 3 units
• Side B (Adjacent to Angle A) = 4 units

Using the Tangent ratio:

tan(A) = Opposite / Adjacent = 3 / 4 = 0.75

A = arctan(0.75) ≈ 36.87°

Since A + B = 90°:

B = 90° - 36.87° = 53.13°

The hypotenuse would be sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5 units.

Example 2: Given Opposite Side and Hypotenuse

Consider a right triangle with:

• Side A (Opposite Angle A) = 5 units
• Hypotenuse = 13 units

Using the Sine ratio:

sin(A) = Opposite / Hypotenuse = 5 / 13 ≈ 0.3846

A = arcsin(0.3846) ≈ 22.62°

Since A + B = 90°:

B = 90° - 22.62° = 67.38°

The adjacent side (Side B) would be sqrt(13² - 5²) = sqrt(169 - 25) = sqrt(144) = 12 units.

This calculator is a handy tool for students, engineers, architects, and anyone needing to quickly determine the angles of a right triangle based on its side lengths.