Pythagorean Theorem Calculator

Compute the hypotenuse or a missing leg of a right triangle, verify triples, and connect geometry to distance and coordinates.

Find Hypotenuse (c)

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Find Missing Leg

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Check Pythagorean Triple

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Pythagorean Theorem: Geometry, Formulas, and Applications

The Pythagorean theorem links the sides of a right triangle and appears in geometry, physics, maps, and coordinate distance problems.

This guide follows your other calculator pages, with boxes, formulas, examples, tables, pitfalls, and practice focused on right triangles and distances.

What Is the Pythagorean Theorem?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs.

Standard form

a² + b² = c², where c is the hypotenuse (side opposite the right angle), and a and b are the legs.

  • Right triangle: One angle is 90 degrees.
  • Hypotenuse: Longest side, opposite the right angle.
  • Legs: The two sides that form the right angle.
a b c Comment
3 4 5 Classic Pythagorean triple
5 12 13 Integer triple
8 15 17 Integer triple
1 1 √2 Isosceles right triangle
Example:
If a = 6 and b = 8, then c² = 6² + 8² = 36 + 64 = 100 → c = 10.

Solving for Hypotenuse and Legs

The calculator cards use rearrangements of a² + b² = c² to find missing sides in right triangles.

Hypotenuse c

c = √(a² + b²)

Example 1: Find c
a = 9, b = 12 → c = √(9² + 12²) = √(81 + 144) = √225 = 15.
Leg a or b

a = √(c² − b²) or b = √(c² − a²), for c greater than the known leg.

Example 2: Find leg
c = 13, a = 5 → b = √(13² − 5²) = √(169 − 25) = √144 = 12.
Known Unknown Formula
a, b c c = √(a² + b²)
b, c a a = √(c² − b²)
a, c b b = √(c² − a²)

Pythagorean Triples

Pythagorean triples are sets of three positive integers (a, b, c) that exactly satisfy a² + b² = c².

Common triples
  • (3, 4, 5)
  • (5, 12, 13)
  • (7, 24, 25)
  • (8, 15, 17)
Example 1: Check (9, 40, 41)
9² + 40² = 81 + 1600 = 1681; 41² = 1681 → triple confirmed.
Example 2: Check (6, 8, 11)
6² + 8² = 36 + 64 = 100; 11² = 121 → not equal, so not a triple.
a b c a² + b² c² Triple?
3 4 5 25 25 Yes
6 8 10 100 100 Yes
4 5 6 41 36 No

Distance and Coordinate Geometry

The distance formula between two points in the plane is just the Pythagorean theorem applied to horizontal and vertical differences.

Distance formula

For points (x₁, y₁) and (x₂, y₂):
Distance d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Example 1:
From (1, 2) to (5, 7):
Δx = 4, Δy = 5 → d = √(4² + 5²) = √(16 + 25) = √41.
Example 2:
From (−3, 4) to (1, 1):
Δx = 4, Δy = −3 → d = √(4² + (−3)²) = √(16 + 9) = 5.
Points Δx Δy Distance
(0,0) to (3,4) 3 4 5
(2,5) to (2,1) 0 −4 4
(−1,−1) to (2,3) 3 4 5

Real-World Applications

The Pythagorean theorem is used wherever straight‑line distances appear: navigation, engineering, physics, and graphics.

📍 Navigation and Maps

Approximates straight‑line distance between two locations given horizontal and vertical offsets.

🏗 Construction and Design

Ensures walls and supports form right angles by checking that side lengths satisfy a² + b² = c².

🎮 Computer Graphics

Used to compute distances between points on the screen and to detect collisions and ranges.

⚙️ Physics and Engineering

Combines perpendicular components of vectors, like velocity or force, into a resultant magnitude.

📐 Robotics and Motion

Helps compute diagonal movements and shortest paths for robot navigation on grids.

📊 Data and Analytics

Forms the basis of Euclidean distance between vectors in many algorithms and machine‑learning models.

Common Mistakes and How to Avoid Them

Most Pythagorean errors come from misidentifying the hypotenuse, negative under the square root, or using the theorem on non‑right triangles.

❌ MISTAKE 1: Wrong side as hypotenuse
Using a shorter side as c breaks the formula c² = a² + b².
✅ Always choose c as the longest side, opposite the right angle.
❌ MISTAKE 2: Negative inside the square root
If c² − a² or c² − b² is negative, the triangle data is impossible for a right triangle.
✅ Check that hypotenuse is larger than either leg before solving for a missing leg.
❌ MISTAKE 3: Using theorem on non‑right triangles
a² + b² = c² only holds when the angle between a and b is 90°.
✅ Confirm that the triangle is right‑angled before applying the theorem.
❌ MISTAKE 4: Squaring and square‑root errors
Forgetting to square both legs or mis‑taking square roots leads to incorrect results.
✅ Perform squaring and root operations step‑by‑step or use a calculator like this page.

Practice Problems with Solutions

Use these problems to drill the theorem; then verify your answers using the calculator cards above.

Basic Level
Problem 1: In a right triangle, a = 5 and b = 12. Find c.
Show Solution

c² = 5² + 12² = 25 + 144 = 169 → c = 13.

Problem 2: A right triangle has hypotenuse 10 and one leg 6. Find the other leg.
Show Solution

b² = 10² − 6² = 100 − 36 = 64 → b = 8.

Intermediate Level
Problem 3: A ladder leans against a wall. The bottom is 6 m from the wall, and the ladder is 10 m long. How high up the wall does it reach?
Show Solution

Height² = 10² − 6² = 100 − 36 = 64 → height = 8 m.

Problem 4: Points A(2, 3) and B(7, 11). Find the distance AB.
Show Solution

Δx = 7 − 2 = 5, Δy = 11 − 3 = 8; d = √(5² + 8²) = √(25 + 64) = √89.

Advanced Level
Problem 5: Show that (9, 12, 15) is a Pythagorean triple and describe how it relates to (3, 4, 5).
Show Solution

9² + 12² = 81 + 144 = 225; 15² = 225 → triple confirmed.
(9, 12, 15) = 3·(3, 4, 5), a scaled version of the basic triple.

Problem 6: A right triangle has area 30 and one leg 10. Find the hypotenuse.
Show Solution

Area = (1/2)ab = 30 ⇒ ab = 60. If a = 10, then b = 60/10 = 6.
c² = 10² + 6² = 100 + 36 = 136 → c = √136 ≈ 11.66.