Explanation
We can use the distance formula in \(2D\) to find the radius of a circle when we know:
- the center of the circle
- a point on the circle
The radius is the distance from the center of the circle to any point on the circle.
Once we find the radius, we can use it to calculate the area of the circle.
Formula / Rule
If the center of the circle is:
$$C=(x_1,y_1)$$
and a point on the circle is:
$$P=(x_2,y_2)$$
then the radius is the distance between these two points:
$$r=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Once we know the radius, the area of the circle is:
$$A=\pi r^2$$
Where:
- \(r\) is the radius of the circle.
- \(A\) is the area of the circle.
- \(\pi\) is approximately:
$$\pi\approx3.1416$$
Remember:
- Every point on a circle is the same distance from its center.
- This distance is the radius.
- Area is measured in square units.
Example
Calculate the area of a circle with center:
$$C=(-1,1)$$
that passes through:
$$P=(3,4)$$
Step 1: Find the Radius
Use the distance formula:
$$r=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Substitute the coordinates:
$$r=\sqrt{(3-(-1))^2+(4-1)^2}$$
Simplify:
\begin{align}
A&=\sqrt{4^2+3^2}\\
\\
&=\sqrt{16+9}\\
\\
&=\sqrt{25}\\
\\
&=5
\end{align}
So, the radius is:
$$r=5\text{ units}$$
Step 2: Find the Area
Use the area formula:
$$A=\pi r^2$$
Substitute:
\begin{align}
A&=\pi(5)^2\\
\\
&=25\pi
\end{align}
So, the area of the circle is:
$$A=25\pi\text{ square units}$$
Using:
$$\pi\approx3.1416$$
the area is approximately:
$$A\approx78.54\text{ square units}$$
Video Explanation
Practice
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 (Current lesson)