Pythagorean Calculator

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Pythagorean Calculator
Find Hypotenuse (c)Find Side (a)Find Side (b)
Answer:
function updateLabels(){var type=document.getElementById('calc_type').value;var l1=document.getElementById('label1');var l2=document.getElementById('label2′);if(type=='hypotenuse'){l1.innerHTML='Side a (length):';l2.innerHTML='Side b (length):';}else if(type=='side_a'){l1.innerHTML='Side b (length):';l2.innerHTML='Hypotenuse c:';}else{l1.innerHTML='Side a (length):';l2.innerHTML='Hypotenuse c:';}document.getElementById('answer').style.display='none';}function calculateResult(){var type=document.getElementById('calc_type').value;var v1=parseFloat(document.getElementById('input1').value);var v2=parseFloat(document.getElementById('input2').value);var showSteps=document.getElementById('steps').checked;if(isNaN(v1)||isNaN(v2)){alert('Please enter valid numeric values for both sides.');return;}var result=0;var details="";if(type=='hypotenuse'){result=Math.sqrt(Math.pow(v1,2)+Math.pow(v2,2));details="Formula: c = \u221A(a\u00B2 + b\u00B2)
c = \u221A("+v1+"\u00B2 + "+v2+"\u00B2)
c = \u221A("+(v1*v1).toFixed(2)+" + "+(v2*v2).toFixed(2)+")
c = \u221A("+(v1*v1+v2*v2).toFixed(2)+")
c = "+result.toFixed(4);}else if(type=='side_a'){if(v2<=v1){alert('Hypotenuse c must be greater than Side b.');return;}result=Math.sqrt(Math.pow(v2,2)-Math.pow(v1,2));details="Formula: a = \u221A(c\u00B2 – b\u00B2)
a = \u221A("+v2+"\u00B2 – "+v1+"\u00B2)
a = \u221A("+(v2*v2).toFixed(2)+" – "+(v1*v1).toFixed(2)+")
a = \u221A("+(v2*v2-v1*v1).toFixed(2)+")
a = "+result.toFixed(4);}else{if(v2<=v1){alert('Hypotenuse c must be greater than Side a.');return;}result=Math.sqrt(Math.pow(v2,2)-Math.pow(v1,2));details="Formula: b = \u221A(c\u00B2 – a\u00B2)
b = \u221A("+v2+"\u00B2 – "+v1+"\u00B2)
b = \u221A("+(v2*v2).toFixed(2)+" – "+(v1*v1).toFixed(2)+")
b = \u221A("+(v2*v2-v1*v1).toFixed(2)+")
b = "+result.toFixed(4);}document.getElementById('finalResult').innerHTML="Result: "+result.toFixed(4);document.getElementById('stepDetails').innerHTML=showSteps?details:"Steps hidden. Check 'Show Solution Steps' to see work.";document.getElementById('answer').style.display='block';}function resetCalc(){document.getElementById('answer').style.display='none';}

Using the Pythagorean Calculator

The pythagorean calculator is a specialized tool designed to solve for any missing side of a right-angled triangle. Based on the fundamental geometric principle known as the Pythagorean Theorem, this tool is essential for students, architects, engineers, and DIY enthusiasts. Whether you need to find the diagonal length of a room or the height of a ladder leaning against a wall, this calculator provides instant, accurate results.

To use this calculator, simply select the side you are trying to find (Hypotenuse, Side a, or Side b) from the dropdown menu, enter the lengths of the two known sides, and click calculate.

Hypotenuse (c)
The longest side of a right triangle, located directly opposite the 90-degree angle.
Side a & Side b
The two shorter sides (legs) of the triangle that form the right angle.

How the Pythagorean Theorem Works

The Pythagorean Theorem states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This relationship is expressed by the famous formula:

a² + b² = c²

Depending on which side you are solving for, the pythagorean calculator rearranges this formula as follows:

  • To find the Hypotenuse (c): c = √(a² + b²)
  • To find Side a: a = √(c² – b²)
  • To find Side b: b = √(c² – a²)

Calculation Examples

Example 1: Finding the Hypotenuse
Imagine you have a right triangle where Side a is 3 cm and Side b is 4 cm. What is the length of the hypotenuse?

  1. Identify values: a = 3, b = 4
  2. Apply formula: c = √(3² + 4²)
  3. Square the numbers: c = √(9 + 16)
  4. Add values: c = √(25)
  5. Result: c = 5 cm

Example 2: Finding a Missing Leg
You have a triangle with a hypotenuse (c) of 13 inches and one side (a) of 5 inches. Find the length of Side b.

  1. Identify values: c = 13, a = 5
  2. Apply formula: b = √(13² – 5²)
  3. Square the numbers: b = √(169 – 25)
  4. Subtract values: b = √(144)
  5. Result: b = 12 inches

Common Questions

Does this work for all triangles?

No, the Pythagorean Theorem and this pythagorean calculator only work for right-angled triangles (triangles that have one angle exactly equal to 90 degrees). For other types of triangles, you would need to use the Law of Cosines or the Law of Sines.

What are Pythagorean Triples?

Pythagorean triples are sets of three positive integers (a, b, c) that perfectly satisfy the formula a² + b² = c². The most famous example is (3, 4, 5). Others include (5, 12, 13) and (8, 15, 17). These are often used in construction to ensure corners are perfectly square.

Why is the result sometimes a decimal?

In most cases, the square root of the sum (or difference) of two squares will not be a whole number. These are called irrational numbers. Our pythagorean calculator provides results rounded to four decimal places for high precision in technical projects.

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