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Right Triangle Visualization
Calculation Steps
Step 1: Square both legs
a² = 3² = 9.0000
b² = 4² = 16.0000
Step 2: Add the squares
a² + b² = 9.0000 + 16.0000 = 25.0000
Step 3: Take the square root
c = √25.0000 = 5.0000
Triangle Properties
6.00
Area
12.00
Perimeter
36.9°
Angle A
53.1°
Angle B
Note: Angle A is opposite side a, Angle B is opposite side b, and the right angle (90°) is opposite the hypotenuse c.
Common Pythagorean Triples
Click any triple to use those values in the calculator.
Result
Hypotenuse c
5.0000
Pythagorean Triple!
3.00² + 4.00² = 5.00²
9.00 + 16.00 = 25.00
Common Triangle Examples
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Practice Problems
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Practice with 3 problems to test your understanding.
?How to Use the Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, a2 + b2 = c2, where c is the hypotenuse (longest side, opposite the right angle) and a and b are the two legs. To find the hypotenuse: c = sqrt(a2 + b2). To find a leg: a = sqrt(c2 - b2). Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17.
What is the Pythagorean Theorem?
The Pythagorean theorem is a fundamental principle in geometry stating that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. This relationship, expressed as a2 + b2 = c2, is used in construction, navigation, surveying, and countless mathematical applications.
Key Facts About the Pythagorean Theorem
- Pythagorean theorem: a squared + b squared = c squared (works only for right triangles)
- c (hypotenuse) is the longest side, opposite the 90-degree angle
- To find hypotenuse: c = sqrt(a squared + b squared)
- To find a leg: a = sqrt(c squared - b squared)
- Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
- Any multiple of a Pythagorean triple is also a Pythagorean triple (6-8-10, 9-12-15)
- The distance formula d = sqrt((x2-x1)2 + (y2-y1)2) is based on the Pythagorean theorem
- Named after ancient Greek mathematician Pythagoras (c. 570-495 BC)
Quick Answer
The Pythagorean theorem states that in a right triangle, a2 + b2 = c2, where c is the hypotenuse (longest side, opposite the right angle) and a and b are the two legs. To find the hypotenuse: c = sqrt(a2 + b2). To find a leg: a = sqrt(c2 - b2). Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17.
Frequently Asked Questions
What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides. Written as: a² + b² = c², where c is the hypotenuse.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Multiples of triples are also triples: (6, 8, 10).
How do I find the hypotenuse?
To find the hypotenuse (c), use the formula c = √(a² + b²). Square both legs, add them together, and take the square root. For example, with legs 3 and 4: c = √(9 + 16) = √25 = 5.
How do I find a leg when I know the hypotenuse?
To find a leg (a) when you know the other leg (b) and hypotenuse (c): a = √(c² - b²). Subtract the square of the known leg from the square of the hypotenuse, then take the square root.
What are common applications of the Pythagorean theorem?
Common uses include: calculating distances on maps, determining diagonal screen sizes, construction and carpentry (ensuring right angles), navigation, physics problems, and computer graphics. Any problem involving right triangles can use this theorem.
Last updated: 2026-01-15
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How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
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