On this page, we practice using the distance formula to find the distance between two points on the coordinate plane.

We practice:

• finding a diagonal distance
• finding a horizontal distance
• finding a vertical distance
• working with negative coordinates
• finding a distance involving a square root

Remember

For two points:

$$P_1=(x_1,y_1)$$

and

$$P_2=(x_2,y_2)$$

use:

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

Example 1 – Diagonal Distance

Find the distance between:

$$P_1=(2,1)$$

and

$$P_2=(5,5)$$

Solution:

\begin{align}
d &= \sqrt{(5-2)^2+(5-1)^2}\\
\\
&= \sqrt{3^2+4^2}\\
\\
&= \sqrt{9+16}\\
\\
&=\sqrt{25}\\
\\
&=5
\end{align}

So, the distance is \(5\) units.

Example 2 – Horizontal Distance

Find the distance between:

$$P_1=(2,3)$$

and

$$P_2=(8,3)$$

Solution:

\begin{align}
d &= \sqrt{(8-2)^2+(3-3)^2}\\
\\
&= \sqrt{6^2+0^2}\\
\\
&= \sqrt{36}\\
\\
&= 6\\
\end{align}

So, the distance is \(6\) units.

Because both points have the same y-coordinate, there is no vertical change.

Example 3 – Vertical Distance

Find the distance between:

$$P_1=(-2,1)$$

and

$$P_2=(-2,7)$$

Solution:

\begin{align}
d &= \sqrt{(-2-(-2))^2+(7-1)^2}\\
\\
&= \sqrt{0^2+6^2}\\
\\
&= \sqrt{36}\\
\\
&= 6\\
\end{align}

So, the distance is \(6\) units.

Because both points have the same x-coordinate, there is no horizontal change.

Example 4 – Negative Coordinates

Find the distance between:

$$P_1=(-3,-2)$$

and

$$P_2=(1,1)$$

Solution:

\begin{align}
d &= \sqrt{(1-(-3))^2+(1-(-2))^2}\\
\\
&= \sqrt{4^2+3^2}\\
\\
&= \sqrt{16+9}\\
\\
&= \sqrt{25}\\
\\
&= 5
\end{align}

So, the distance is \(5\) units.

Example 5 – Answer in Square-Root Form

Find the distance between:

$$P_1=(1,1)$$

and

$$P_2=(3,4)$$

Solution:

\begin{align}
d &= \sqrt{(3-1)^2+(4-1)^2}\\
\\
&= \sqrt{2^2+3^2}\\
\\
&= \sqrt{4+9}\\
\\
&= \sqrt{13}\\
\end{align}

As a decimal:

$$d\approx3.61$$

So, the distance is:

$$\sqrt{13}\approx3.61\text{ units}$$

Practice

• More Examples
• Take the Quiz

Continue Learning

1. Plotting Points on the Coordinate Plane (\(2D\))
2. Which Quadrant or Axis? Points on the Coordinate Plane \(2D\)
3. Distance Formula in \(2D\)
4. Distance between Two Points in \(2D\)
5. Distance Formula: Radius and Area of a Circle

Navigation

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