Pythagorean Theorem

Pythagorean Theorem - Solve mathematical problems with step-by-step solutions.

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Pythagorean Theorem Calculator

Solve right triangle problems using the Pythagorean theorem: a² + b² = c²

What to Solve

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Common Examples

Pythagorean Theorem

In a right triangle with legs a and b, and hypotenuse c: a² + b² = c²

How the Pythagorean Theorem Calculator Works

The Pythagorean Theorem Calculator solves problems involving right triangles using one of the most famous mathematical formulas in history. Named after the ancient Greek mathematician Pythagoras, this theorem establishes a fundamental relationship between the three sides of a right triangle. It states that the square of the hypotenuse (longest side) equals the sum of squares of the other two sides.

The Pythagorean Theorem

Standard Form: a2 + b2 = c2

Solving for Hypotenuse: c = √(a2 + b2)

Solving for a Leg: a = √(c2 - b2) or b = √(c2 - a2)

Where 'a' and 'b' are the lengths of the two legs (sides forming the right angle), and 'c' is the length of the hypotenuse (side opposite the right angle).

Key Concepts

  • Right Triangle Only: This theorem ONLY applies to triangles with a 90-degree angle
  • Hypotenuse: Always the longest side, opposite the right angle
  • Legs: The two sides that form the right angle (perpendicular to each other)
  • Pythagorean Triples: Integer sets that satisfy the theorem (3-4-5, 5-12-13, 8-15-17, etc.)
  • Geometric Proof: The area of the square on the hypotenuse equals the sum of areas of squares on the legs
  • Applications: Distance calculations, construction, navigation, and countless real-world problems

Detailed Examples

Example 1: Finding the Hypotenuse (Classic 3-4-5 Triangle)

Given: Right triangle with legs a = 3 feet and b = 4 feet

Find: Hypotenuse c

Formula: c = √(a2 + b2)

Calculation: c = √(32 + 42) = √(9 + 16) = √25 = 5 feet

Answer: The hypotenuse is 5 feet (a Pythagorean triple)

Example 2: Finding a Leg

Given: Hypotenuse c = 13 cm, one leg a = 5 cm

Find: Other leg b

Formula: b = √(c2 - a2)

Calculation: b = √(132 - 52) = √(169 - 25) = √144 = 12 cm

Answer: The other leg is 12 cm (5-12-13 Pythagorean triple)

Example 3: Ladder Against a Wall

Problem: A 10-meter ladder leans against a wall. The base is 6 meters from the wall. How high does it reach?

Setup: Hypotenuse (ladder) c = 10 m, base (distance from wall) a = 6 m

Find: Height on wall b

Calculation: b = √(102 - 62) = √(100 - 36) = √64 = 8 meters

Answer: The ladder reaches 8 meters up the wall

Example 4: Diagonal of a Rectangle

Problem: Find the diagonal of a rectangle 9 inches wide and 12 inches tall

Setup: Width a = 9 in, height b = 12 in, diagonal = c

Calculation: c = √(92 + 122) = √(81 + 144) = √225 = 15 inches

Answer: The diagonal measures 15 inches

Example 5: Distance Across a Park

Problem: To cross a rectangular park, you can walk 80m north then 60m east, or cut diagonally. How much shorter is the diagonal path?

Rectangular path: 80 + 60 = 140 meters

Diagonal: c = √(802 + 602) = √(6400 + 3600) = √10,000 = 100 meters

Savings: 140 - 100 = 40 meters shorter via diagonal

Example 6: Non-Integer Result

Given: Right triangle with legs a = 5 cm and b = 7 cm

Calculation: c = √(52 + 72) = √(25 + 49) = √74 ≈ 8.60 cm

Note: Not all right triangles have integer side lengths—decimal results are common

Tips and Best Practices

Essential Tips for Pythagorean Theorem

  • Identify the Right Angle: Only use this theorem for right triangles (90° angle present)
  • Hypotenuse is Longest: The hypotenuse (c) is always the longest side, opposite the right angle
  • Square First, Then Add: Calculate a2 and b2 separately, add them, then take the square root
  • Label Carefully: When finding a leg, make sure you subtract the KNOWN leg from the hypotenuse
  • Check Your Result: The hypotenuse must be longer than either leg. If not, recheck your work
  • Memorize Common Triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples (6-8-10, 9-12-15, etc.)
  • Unit Consistency: All sides must use the same units before calculating
  • Perfect Squares: Recognize perfect squares (4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225) for faster mental math
  • Verification: Plug your answer back into a2 + b2 = c2 to verify it works
  • Negative Impossible: If you get a negative under the square root, you've made an error (likely swapped c with a leg)

Pythagorean Triples

Common Pythagorean Triples

Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a2 + b2 = c2. Memorizing these can speed up calculations:

Primitive Triples: (where a, b, c share no common factor)

  • 3-4-5 (most famous)
  • 5-12-13
  • 8-15-17
  • 7-24-25
  • 20-21-29
  • 9-40-41
  • 12-35-37
  • 11-60-61

Scaled Triples: Any multiple of a Pythagorean triple is also a triple

  • 6-8-10 (2 × 3-4-5)
  • 9-12-15 (3 × 3-4-5)
  • 10-24-26 (2 × 5-12-13)
  • 15-20-25 (5 × 3-4-5)

Real-World Applications

Real-World Applications

  • Construction: Ensuring corners are square, calculating rafter lengths, and determining diagonal bracing
  • Navigation: Computing direct distances, shortest routes, and triangulation
  • Surveying: Measuring inaccessible distances using right triangle relationships
  • Architecture: Designing stairs, ramps, roofs, and ensuring structural squareness
  • Engineering: Force analysis, bridge design, and structural calculations
  • Computer Graphics: Distance calculations, collision detection, and 2D/3D positioning
  • Sports: Baseball diamond distances, field measurements, and optimal angles
  • Physics: Vector components, projectile motion, and force decomposition
  • Woodworking: Cutting accurate angles and ensuring square assemblies
  • Landscaping: Plotting diagonal paths, fence lines, and garden layouts

Frequently Asked Questions