Calculate missing side lengths of right triangles using the Pythagorean theorem a² + b² = c². Enter any two values to find the third side with step-by-step solutions and visual proof.
The Pythagorean theorem is one of the most fundamental principles in mathematics, establishing a relationship between the sides of a right triangle. Named after the ancient Greek mathematician Pythagoras, this theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides.
This mathematical relationship has applications in construction, navigation, engineering, computer graphics, and countless other fields where distance calculations and right triangle properties are important. Understanding the Pythagorean theorem is essential for geometry, trigonometry, and advanced mathematics.
Where a and b are the legs, and c is the hypotenuse
When you know b and c, find a
When you know a and c, find b
When you know both legs a and b, find c
Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem. These combinations are particularly useful in construction and design work because they create perfect right triangles with whole number measurements.
| a | b | c | Type | Common Uses |
|---|---|---|---|---|
| 3 | 4 | 5 | Primitive | Basic construction, education |
| 5 | 12 | 13 | Primitive | Surveying, carpentry |
| 8 | 15 | 17 | Primitive | Architecture, engineering |
| 7 | 24 | 25 | Primitive | Large construction projects |
| 6 | 8 | 10 | 3-4-5 Multiple | Small-scale building |
| 9 | 12 | 15 | 3-4-5 Multiple | Room layouts, framing |
| 10 | 24 | 26 | 5-12-13 Multiple | Property boundaries |
Construction and Carpentry: Ensuring corners are perfectly square (90 degrees) by using the 3-4-5 rule or other Pythagorean triples. This is essential for building foundations, framing, and laying out rooms.
Navigation and GPS: Calculating straight-line distances between points when you know the horizontal and vertical separations. This is fundamental to GPS systems and mapping applications.
Engineering and Design: Determining diagonal braces for structural support, calculating cable lengths for suspension systems, and designing triangular trusses for bridges and buildings.
Computer Graphics: Calculating distances between pixels, determining collision detection in games, and creating 3D graphics where distance calculations are crucial.
Step 1: Identify the Triangle Type - Confirm you have a right triangle with one 90-degree angle. The Pythagorean theorem only applies to right triangles.
Step 2: Identify Known and Unknown Sides - Determine which two sides you know and which one you need to find. Remember that the hypotenuse is always the longest side.
Step 3: Choose the Correct Formula - Use the appropriate variation of a² + b² = c² based on which side you're solving for.
Step 4: Substitute and Calculate - Plug in your known values and solve for the unknown. Don't forget to take the square root when solving for a side length.
Step 5: Verify Your Answer - Check that your result makes sense (hypotenuse should be longest) and verify by substituting all values back into the original theorem.
Forgetting the Square Root: When solving for a side length, remember to take the square root of both sides of the equation. The theorem gives you the square of the side length.
Misidentifying the Hypotenuse: Always remember that the hypotenuse is the side opposite the right angle and is always the longest side of the triangle.
Using Wrong Triangle Type: The Pythagorean theorem only works for right triangles. For other triangles, use the law of cosines or other geometric relationships.
Unit Inconsistency: Ensure all measurements use the same units before calculating. Convert units as necessary to maintain consistency.
Ancient Origins: While named after Pythagoras (c. 570-495 BCE), the relationship was known to Babylonians over 1000 years earlier. The theorem appears in mathematical texts from many ancient civilizations.
Geometric Proof: The classic proof involves arranging four identical right triangles to form a square, demonstrating that the area relationships match the algebraic formula.
Algebraic Proof: Modern proofs use algebraic manipulation to show that the relationship holds for all right triangles, regardless of size or orientation.
Practical Applications in History: Ancient builders used the 3-4-5 rope trick to create right angles for construction, demonstrating the practical importance of this mathematical principle.
3D Pythagorean Theorem: For three-dimensional space, the distance formula d = √(x² + y² + z²) extends the Pythagorean concept to find distances in 3D coordinates.
Law of Cosines: This generalizes the Pythagorean theorem for all triangles: c² = a² + b² - 2ab cos(C), where C is the angle opposite side c.
Trigonometric Relationships: The Pythagorean theorem is fundamental to trigonometric identities like sin²θ + cos²θ = 1.