'+a+'² + '+b+'² = c²
'+(Math.pow(a,2))+' + '+(Math.pow(b,2))+' = c²
'+(Math.pow(a,2)+Math.pow(b,2))+' = c²
c = √'+(Math.pow(a,2)+Math.pow(b,2))+'
c = '+result.toFixed(4)+'';}else if(type==='a'){if(isNaN(b)||isNaN(c)){alert('Please enter values for side b and side c');return;}if(c<=b){alert('Hypotenuse (c) must be longer than leg (b)');return;}result=Math.sqrt(Math.pow(c,2)-Math.pow(b,2));stepsHtml='Using the formula: a² + b² = c²
a² + '+b+'² = '+c+'²
a² + '+(Math.pow(b,2))+' = '+(Math.pow(c,2))+'
a² = '+(Math.pow(c,2))+' – '+(Math.pow(b,2))+'
a² = '+(Math.pow(c,2)-Math.pow(b,2))+'
a = √'+(Math.pow(c,2)-Math.pow(b,2))+'
a = '+result.toFixed(4)+'';}else if(type==='b'){if(isNaN(a)||isNaN(c)){alert('Please enter values for side a and side c');return;}if(c<=a){alert('Hypotenuse (c) must be longer than leg (a)');return;}result=Math.sqrt(Math.pow(c,2)-Math.pow(a,2));stepsHtml='Using the formula: a² + b² = c²
'+a+'² + b² = '+c+'²
'+(Math.pow(a,2))+' + b² = '+(Math.pow(c,2))+'
b² = '+(Math.pow(c,2))+' – '+(Math.pow(a,2))+'
b² = '+(Math.pow(c,2)-Math.pow(a,2))+'
b = √'+(Math.pow(c,2)-Math.pow(a,2))+'
b = '+result.toFixed(4)+'';}document.getElementById('calculatorAnswer').style.display='block';document.getElementById('answerDisplay').innerHTML=showSteps?stepsHtml:'Result: '+result.toFixed(4)+'';}function resetCalc(){document.getElementById('calculatorAnswer').style.display='none';document.getElementById('calc_type').value='c';toggleInputs();}
Calculator Use
This pythagorean theorem calculator is a professional-grade geometry tool designed to solve for any missing side of a right-angled triangle. Whether you need to find the hypotenuse (the longest side) or one of the shorter legs, this tool provides instant results based on the fundamental principles of Euclidean geometry.
To use the calculator, simply select which side you are trying to find, enter the two known lengths, and click calculate. Our tool handles decimals and large numbers with precision, making it ideal for students, engineers, and construction professionals alike.
- Side a (Leg)
- One of the two sides that form the 90-degree right angle.
- Side b (Leg)
- The second side that forms the right angle, often representing the base or height.
- Side c (Hypotenuse)
- The longest side of the right triangle, located directly opposite the right angle.
How It Works
The Pythagorean Theorem is one of the most famous equations in mathematics. It states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The mathematical formula is expressed as:
a² + b² = c²
To find specific sides, the formula can be rearranged:
- To find the Hypotenuse (c): c = √(a² + b²)
- To find a Leg (a): a = √(c² – b²)
- To find a Leg (b): b = √(c² – a²)
Calculation Example
Example: Suppose you are building a wooden ramp. The height of the ramp (Side a) is 3 feet, and the horizontal distance along the ground (Side b) is 4 feet. How long must the ramp board (Hypotenuse c) be?
Step-by-step solution using the pythagorean theorem calculator:
- Identify the sides: a = 3, b = 4.
- Square both numbers: 3² = 9 and 4² = 16.
- Add the squares together: 9 + 16 = 25.
- Take the square root of the sum: √25 = 5.
- Result: The ramp board must be 5 feet long.
The Importance of Right Triangles
It is critical to remember that this pythagorean theorem calculator only works for right-angled triangles (triangles where one angle is exactly 90 degrees). If you are dealing with an acute or obtuse triangle, you must use the Law of Cosines instead. The Pythagorean Theorem is actually a special case of the Law of Cosines where the angle is 90 degrees and the cosine value is zero.
Real World Applications
While it may seem like a purely academic exercise, the Pythagorean Theorem is used daily in many fields:
- Construction & Architecture: Ensuring that the corners of buildings are perfectly square by using the "3-4-5 rule."
- Navigation: Calculating the shortest distance (as the crow flies) between two points on a map.
- Screen Sizes: TV and monitor sizes are sold by their diagonal length, which is calculated using the width and height of the screen.
- Emergency Services: Firefighters use the theorem to determine how far from a building the base of a ladder needs to be to reach a specific window.
Common Questions
What is a Pythagorean Triple?
A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². The most famous example is (3, 4, 5). Others include (5, 12, 13) and (8, 15, 17). These are useful for quick mental math in construction and design.
Can the theorem be used for 3D shapes?
Yes, there is a 3D version of the theorem used to find the diagonal of a rectangular prism: d² = l² + w² + h². Our pythagorean theorem calculator focuses on 2D space, but the logic remains a foundation for higher-dimensional mathematics.
Who actually discovered the theorem?
While named after the Greek mathematician Pythagoras, historical evidence suggests that Babylonian and Indian mathematicians were using the principles of the theorem centuries before he lived. Pythagoras is credited with bringing the knowledge to the Mediterranean world and potentially providing the first formal proof.