Right-Angle Triangle Calculator
Enter any two known values (sides or an acute angle) to calculate the remaining sides and angles of a right-angled triangle.
Calculated Triangle Properties:
'; resultDiv.innerHTML += 'Side Opposite (o): ' + calculatedSideO.toFixed(4) + "; resultDiv.innerHTML += 'Side Adjacent (a): ' + calculatedSideA.toFixed(4) + "; resultDiv.innerHTML += 'Hypotenuse (h): ' + calculatedHypotenuse.toFixed(4) + "; resultDiv.innerHTML += 'Angle A: ' + calculatedAngleA.toFixed(4) + ' degrees'; resultDiv.innerHTML += 'Angle B: ' + calculatedAngleB.toFixed(4) + ' degrees'; resultDiv.innerHTML += 'Angle C (Right Angle): 90 degrees'; } function clearTrigInputs() { document.getElementById('sideOpposite').value = "; document.getElementById('sideAdjacent').value = "; document.getElementById('hypotenuse').value = "; document.getElementById('angleA').value = "; document.getElementById('trigResult').innerHTML = "; }Understanding Right-Angle Triangles and Trigonometry
A right-angle triangle is a fundamental shape in geometry, characterized by one angle measuring exactly 90 degrees. The sides of a right-angle triangle have specific names relative to one of the other two acute angles (angles less than 90 degrees):
- Hypotenuse (h): The longest side, always opposite the 90-degree angle.
- Opposite (o): The side directly across from the acute angle you are considering.
- Adjacent (a): The side next to the acute angle you are considering, not the hypotenuse.
The Pythagorean Theorem
One of the most well-known properties of a right-angle triangle is the Pythagorean Theorem, which states that the square of the hypotenuse (h) is equal to the sum of the squares of the other two sides (opposite 'o' and adjacent 'a').
h² = o² + a²
This theorem allows you to find the length of any side if you know the lengths of the other two.
Trigonometric Ratios (SOH CAH TOA)
Trigonometry provides a way to relate the angles of a right-angle triangle to the ratios of its sides. The three primary trigonometric ratios are Sine, Cosine, and Tangent, often remembered by the mnemonic SOH CAH TOA:
- SOH: Sine = Opposite / Hypotenuse (
sin(Angle) = o / h) - CAH: Cosine = Adjacent / Hypotenuse (
cos(Angle) = a / h) - TOA: Tangent = Opposite / Adjacent (
tan(Angle) = o / a)
These ratios allow you to find unknown sides if you know an angle and one side, or to find unknown angles if you know two sides.
How to Use the Right-Angle Triangle Calculator
Our Right-Angle Triangle Calculator simplifies these calculations. To use it:
- Identify Known Values: Look at your right-angle triangle and determine which two values you know. This could be two sides, or one side and one acute angle.
- Enter Values: Input your known values into the corresponding fields: "Side Opposite (o)", "Side Adjacent (a)", "Hypotenuse (h)", or "Angle A (degrees)". Make sure to enter at least two values.
- Calculate: Click the "Calculate" button. The calculator will automatically determine the remaining sides and angles using the appropriate trigonometric functions and the Pythagorean theorem.
- Review Results: The results section will display the calculated lengths of the sides and the measures of the acute angles (Angle A and Angle B), along with the fixed 90-degree angle.
Practical Applications
Right-angle triangles and trigonometry are essential in many fields:
- Engineering and Architecture: Calculating forces, structural stability, and dimensions.
- Navigation: Determining distances and bearings (e.g., GPS, aviation).
- Physics: Analyzing vectors, projectile motion, and wave properties.
- Surveying: Measuring land, heights of buildings, and distances.
- Computer Graphics: Rendering 3D objects and calculating perspectives.
Examples
Example 1: Knowing Two Sides (Opposite and Adjacent)
Imagine you have a right-angle triangle where the side opposite Angle A is 3 units, and the side adjacent to Angle A is 4 units.
- Input: Side Opposite (o) = 3, Side Adjacent (a) = 4
- Calculation:
- Hypotenuse (h) = √(3² + 4²) = √(9 + 16) = √25 = 5
- Angle A = arctan(3/4) ≈ 36.8699 degrees
- Angle B = 90 – 36.8699 = 53.1301 degrees
- Output: Hypotenuse = 5.0000, Angle A ≈ 36.8699°, Angle B ≈ 53.1301°
Example 2: Knowing Hypotenuse and One Angle
Suppose the hypotenuse of a right-angle triangle is 10 units, and Angle A is 30 degrees.
- Input: Hypotenuse (h) = 10, Angle A = 30
- Calculation:
- Side Opposite (o) = 10 * sin(30°) = 10 * 0.5 = 5
- Side Adjacent (a) = 10 * cos(30°) = 10 * 0.866025 ≈ 8.6603
- Angle B = 90 – 30 = 60 degrees
- Output: Side Opposite ≈ 5.0000, Side Adjacent ≈ 8.6603, Angle B = 60.0000°
Example 3: Knowing One Side and Hypotenuse
A right-angle triangle has a hypotenuse of 13 units and a side opposite Angle A of 5 units.
- Input: Hypotenuse (h) = 13, Side Opposite (o) = 5
- Calculation:
- Side Adjacent (a) = √(13² – 5²) = √(169 – 25) = √144 = 12
- Angle A = arcsin(5/13) ≈ 22.6199 degrees
- Angle B = 90 – 22.6199 = 67.3801 degrees
- Output: Side Adjacent = 12.0000, Angle A ≈ 22.6199°, Angle B ≈ 67.3801°