Pythagorean Theorem — Squares on Sides

# Pythagorean Theorem — Squares on Sides

## Plain English first

Draw a right triangle. Now grow a square outward from each of its three sides.

The square on the longest side (the hypotenuse) has exactly the same area as the other two squares combined. Every time. For any right triangle, any size.

That's the theorem. It's not just a formula — it's a fact about area. The squares on the two legs, if you cut them up and rearrange the pieces, will tile exactly into the square on the hypotenuse. No gaps, no overlaps.

---

## Standard math notation

```
a² + b² = c²

Where:
a = one leg of the right triangle
b = the other leg
c = hypotenuse (the side opposite the right angle — always the longest)

a² = area of square built on side a
b² = area of square built on side b
c² = area of square built on side c

The theorem says: a² + b² = c²
Solving for c: c = √(a² + b²)
Solving for a: a = √(c² - b²)
```

---

## Verbose Python with descriptive names

```python
def compute_hypotenuse_from_two_legs(
length_of_first_leg,
length_of_second_leg
):
"""
The square on the hypotenuse equals the sum of squares on both legs.
To find the hypotenuse: add the two squared areas, take the square root.
"""
area_of_square_on_first_leg = length_of_first_leg ** 2
area_of_square_on_second_leg = length_of_second_leg ** 2
combined_area_of_both_squares = area_of_square_on_first_leg + area_of_square_on_second_leg
length_of_hypotenuse = combined_area_of_both_squares ** 0.5
return length_of_hypotenuse

hypotenuse = compute_hypotenuse_from_two_legs(
length_of_first_leg=3,
length_of_second_leg=4
)
print(f"Hypotenuse: {hypotenuse}") # 5.0

def compute_missing_leg(
length_of_hypotenuse,
length_of_known_leg
):
"""
If we know the hypotenuse and one leg, the other leg is:
missing_leg = √(hypotenuse² - known_leg²)
"""
area_of_hypotenuse_square = length_of_hypotenuse ** 2
area_of_known_leg_square = length_of_known_leg ** 2
area_remaining = area_of_hypotenuse_square - area_of_known_leg_square
length_of_missing_leg = area_remaining ** 0.5
return length_of_missing_leg

print(compute_missing_leg(
length_of_hypotenuse=5,
length_of_known_leg=3
)) # 4.0
```

---

## Common Pythagorean triples (worth memorizing)

```python
PYTHAGOREAN_TRIPLES = [
(3, 4, 5), # most common
(5, 12, 13),
(8, 15, 17),
(7, 24, 25),
(6, 8, 10), # 3-4-5 scaled by 2
(9, 12, 15), # 3-4-5 scaled by 3
]
# If you see two of these numbers in a problem, the third is free
```

---

## This theorem is everywhere

The Pythagorean theorem is the foundation for:
- **Distance formula** — find distance between two points on a grid
- **Trigonometry** — sin²θ + cos²θ = 1 is Pythagorean on the unit circle
- **3D distance** — d = √(x²+y²+z²) is the theorem applied twice

---

## See also
- [Distance Formula — Pythagorean on a Grid](/articles/distance-formula)
- [Sine and Cosine — Waves from Orbit](/articles/sin-cos-waves)
- [Visual Triangle Geometry](/articles/794e2e02-16a0-43fc-955a-ab27f8d1de8d)
- [Math Foundations — Visual Table of Contents](/articles/d404884f-54fc-4289-b3f1-baaad2bec6b2)