how to find distance between two points

To find the distance between two points, you use the distance formula , which comes from the Pythagorean theorem.

Basic idea (2D)

If your two points are
A(x1,y1)A(x_1,y_1)A(x1​,y1​) and B(x2,y2)B(x_2,y_2)B(x2​,y2​), then the distance between them is:

d=(x2−x1)2+(y2−y1)2d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}d=(x2​−x1​)2+(y2​−y1​)2​

This is just a right triangle: one leg is the horizontal change (x2−x1)(x_2-x_1)(x2​−x1​), the other is the vertical change (y2−y1)(y_2-y_1)(y2​−y1​), and the distance is the hypotenuse.

Step‑by‑step recipe

1. Write down the coordinates
   • Point A: (x1,y1)(x_1,y_1)(x1​,y1​)
   • Point B: (x2,y2)(x_2,y_2)(x2​,y2​)

2. Find the differences
   • Horizontal change: x2−x1x_2-x_1x2​−x1​
   • Vertical change: y2−y1y_2-y_1y2​−y1​

3. Square the differences
   • (x2−x1)2(x_2-x_1)^2(x2​−x1​)2
   • (y2−y1)2(y_2-y_1)^2(y2​−y1​)2

4. Add them together
   • (x2−x1)2+(y2−y1)2(x_2-x_1)^2+(y_2-y_1)^2(x2​−x1​)2+(y2​−y1​)2

5. Take the square root
   • d=(x2−x1)2+(y2−y1)2d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}d=(x2​−x1​)2+(y2​−y1​)2​

Quick example

Find the distance between (3,4)(3,4)(3,4) and (4,3)(4,3)(4,3):

• x_1=3,\y_1=4
• x_2=4,\y_2=3
• x_2-x_1=4-3=1
• y_2-y_1=3-4=-1
• Square: 1^2=1,\(-1)^2=1
• Add: 1+1=2
• Distance: d=\sqrt{2}\approx 1.414

Special cases

• If both points are on a horizontal line (same y): distance is |x_2-x_1|.

• If both points are on a vertical line (same x): distance is |y_2-y_1|.

In 3D (if you ever need it)

For points A(x_1,y_1,z_1) and B(x_2,y_2,z_2), the distance is:

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

TL;DR:
Remember this one line and you’re set:

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}