# Distance Formula — Pythagorean on a Grid
> **Animation coming** — full Sora 2 prompt below. Video will replace this placeholder.
## Plain English first
You have two points on a grid. You want the straight-line distance between them.
Draw a right triangle. Make the horizontal leg the difference in x-coordinates, and the vertical leg the difference in y-coordinates. The straight-line distance is the hypotenuse of that triangle.
This is the Pythagorean theorem — applied to a coordinate grid. The formula looks complicated but it's just a² + b² = c², solved for c, where a = Δx and b = Δy.
---
## Standard math notation
```
Distance formula:
d = √((x₂ − x₁)² + (y₂ − y₁)²)
This is directly the Pythagorean theorem:
horizontal leg = (x₂ − x₁)
vertical leg = (y₂ − y₁)
hypotenuse = d
d² = (x₂−x₁)² + (y₂−y₁)²
d = √(above)
Midpoint formula (center of the segment):
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Average the x-coordinates, average the y-coordinates.
```
---
## Verbose Python with descriptive names
```python
def compute_distance_between_two_points(
x_of_first_point,
y_of_first_point,
x_of_second_point,
y_of_second_point
):
"""
Distance = hypotenuse of the right triangle connecting the two points.
Horizontal leg = difference in x. Vertical leg = difference in y.
Pythagorean theorem: d² = (Δx)² + (Δy)² → d = √(Δx² + Δy²)
"""
horizontal_distance = x_of_second_point - x_of_first_point
vertical_distance = y_of_second_point - y_of_first_point
sum_of_squares = horizontal_distance**2 + vertical_distance**2
straight_line_distance = sum_of_squares ** 0.5 # square root
return straight_line_distance
def compute_midpoint_between_two_points(
x_of_first_point,
y_of_first_point,
x_of_second_point,
y_of_second_point
):
"""
Midpoint is the average of each coordinate separately.
Halfway in x, halfway in y.
"""
x_of_midpoint = (x_of_first_point + x_of_second_point) / 2
y_of_midpoint = (y_of_first_point + y_of_second_point) / 2
return x_of_midpoint, y_of_midpoint
# Classic 3-4-5 triangle on a grid
distance = compute_distance_between_two_points(
x_of_first_point=1,
y_of_first_point=1,
x_of_second_point=4,
y_of_second_point=5
)
print(f"Distance: {distance}") # 5.0 (horizontal=3, vertical=4, hypotenuse=5)
midpoint = compute_midpoint_between_two_points(1, 1, 4, 5)
print(f"Midpoint: {midpoint}") # (2.5, 3.0)
# 3D distance — same idea, add a third leg
def compute_3d_distance(x1, y1, z1, x2, y2, z2):
"""
Pythagorean theorem applied twice: once for x+y, once for result+z.
d = √(Δx² + Δy² + Δz²)
"""
dx, dy, dz = x2-x1, y2-y1, z2-z1
return (dx**2 + dy**2 + dz**2) ** 0.5
```
---
## Sora 2 video prompt
```
8-second animation. Two points A(1,1) and B(4,5) appear on a coordinate grid.
A horizontal dashed line forms between them labeled 'run = 3'. A vertical
dashed line forms labeled 'rise = 4'. The right triangle is complete and its
sides glow. Squares grow on each leg: 9 and 16. They combine: √25 = 5. The
hypotenuse lights up labeled 'd = 5'. Formula morphs into d = √((x₂-x₁)²+(y₂-y₁)²).
Warm earth tones, clean coordinate grid.
```
---
## Builds on
- [Pythagorean Theorem — Squares on Sides](/articles/4d64ff4e-28db-4c2b-8415-8751fd5adecf)
- [Slope and Linear Equations — Rise Over Run](/articles/slope-linear)
## See also
- [Sine and Cosine — Waves from a Spinning Point](/articles/sin-cos)
- [Visual Triangle Geometry](/articles/794e2e02-16a0-43fc-955a-ab27f8d1de8d)
- [Math Foundations — Visual Table of Contents](/articles/d404884f-54fc-4289-b3f1-baaad2bec6b2)