On this page, we practice using the distance formula to find the radius and area of different circles.
We practice:
- finding the radius from two points
- using positive and negative coordinates
- finding the area in terms of \(\pi\)
- finding approximate decimal areas
- recognizing horizontal and vertical radii
Remember
If the center is:
$$C=(x_1,y_1)$$
and a point on the circle is:
$$P=(x_2,y_2)$$
then:
$$r=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Then use:
$$A=\pi r^2$$
Example 1
A circle has center:
$$C=(0,0)$$
and passes through:
$$P=(3,4)$$
Find the radius and area.
Find the radius:
\begin{align}
r &= \sqrt{(3-0)^2+(4-0)^2}\\
\\
&= \sqrt{3^2+4^2}\\
\\
&= \sqrt{9+16}\\
\\
&= 5
\end{align}
Now find the area:
\begin{align}
A &= \pi(5)^2\\
\\
&= 25\pi
\end{align}
So:
$$r=5\text{ units}$$
and
$$A=25\pi\text{ square units}$$
Example 2
A circle has center:
$$C=(2,1)$$
and passes through:
$$P=(8,1)$$
Find the radius and area.
Find the radius:
\begin{align}
r &= \sqrt{(8-2)^2+(1-1)^2}\\
\\
&= \sqrt{6^2+0^2}\\
\\
&= 6
\end{align}
Now find the area:
\begin{align}
A &= \pi(6)^2\\
\\
&= 36\pi
\end{align}
So:
$$r=6\text{ units}$$
and
$$A=36\pi\text{ square units}$$
Example 3
A circle has center:
$$C=(-3,-2)$$
and passes through:
$$P=(-3,3)$$
Find the radius and area.
Find the radius:
\begin{align}
r &= \sqrt{(-3-(-3))^2+(3-(-2))^2}\\
\\
&= \sqrt{0^2+5^2}\\
\\
&= 5
\end{align}
Now find the area:
\begin{align}
A &= \pi(5)^2\\
\\
&= 25\pi
\end{align}
So:
$$r=5\text{ units}$$
and
$$A=25\pi\text{ square units}$$
Example 4
A circle has center:
$$C=(-2,1)$$
and passes through:
$$P=(1,5)$$
Find the radius and area.
Find the radius:
\begin{align}
r &= \sqrt{(1-(-2))^2+(5-1)^2}\\
\\
&= \sqrt{3^2+4^2}\\
\\
&= \sqrt{9+16}\\
\\
&= 5
\end{align}
Now find the area:
\begin{align}
A &= \pi(5)^2\\
\\
&= 25\pi
\end{align}
So:
$$r=5\text{ units}$$
and
$$A=25\pi\text{ square units}$$
Example 5 – Approximate Area
A circle has center:
$$C=(1,2)$$
and passes through:
$$P=(1,6)$$
Find the radius and area.
Find the radius:
\begin{align}
r &= \sqrt{(1-1)^2+(6-2)^2}\\
\\
&= \sqrt{0^2+4^2}\\
\\
&= 4
\end{align}
Now find the area:
\begin{align}
A &= \pi(4)^2\\
\\
&= 16\pi
\end{align}
Using:
$$\pi\approx3.1416$$
we get:
(A\approx50.27)
So:
$$r=4\text{ units}$$
and
$$A=16\pi\approx50.27\text{ square units}$$
Practice
- More Examples (Current page)
- Take the Quiz
Continue Learning
- Plotting Points on the Coordinate Plane (\(2D\))
- Which Quadrant or Axis? Points on the Coordinate Plane \(2D\)
- Distance Formula in \(2D\)
- Distance between Two Points in \(2D\)
- Distance Formula: Radius and Area of a Circle