Knowing how to solve for the area, circumference, diameter, and radius of a circle is an important math concept to understand. Take a look at the examples below to see how each is calculated.
Formulas
Diameter and Radius
The diameter (\(d\)) is the distance straight across a circle. The radius (\(r\)) is the distance from the center of a circle to any point on the circle’s edge.
It’s important to note that the radius is always half the length of the diameter.
- To find the radius from the diameter: \(r=\dfrac{d}{2}\)
- To find the diameter from the radius: \(d=2r\)
Circumference and Area
The circumference (\(C\)) is the distance around the outside of a circle. The area (\(A\)) is the total space inside the circle.
- Circumference formula: \(C=2\pi r\) or \(C=\pi d\)
- Area formula: \(A=\pi r^2\)
Example Problems
Diameter
To solve this, we need to work backward from the area to find the radius.
First, we need to plug our area (49π) into the area formula and solve for \(r\).
\(49\pi =\pi r^2\)
\(\dfrac{49\pi}{\pi}=\dfrac{\pi r^2}{\pi}\)
\(49 = r^2\)
\(r = 7\)
We now know that the radius is 7 feet. Using out formula above (\(d=2r\)), we can use the radius to find the diameter.
\(d=2\times 7=14\)
The diameter of the circle is 14 feet!
Radius
The distance of one full rotation is the same as the circumference. Therefore, we know the circumference of the wheel is 30π inches.
First, we need to plug the circumference of the wheel into the circumference formula and solve for \(r\).
\(30\pi =2 \pi r\)
\(r= \dfrac{30\pi}{2 \pi}=15\)
We now know that the radius is 15 inches!
Area
The area formula requires the radius, so we need to find that first using the diameter.
Now, we can use the radius to find the area.
We now know that the area of the circle is 81π in2!
Circumference
First, we need to find the radius from the area.
\(r^2=100 \rightarrow r=10\text{ yd}\)
Then, use the radius to find the circumference.
We now know that the circumference is 20π yards!
