A circle is a perfectly round geometric shape. You have probably seen many things that have the same shape as a circle, for example a doughnut or a bicycle wheel. Here are a few drawings of circles:
Think About This
Try drawing some circles yourself. It’s a bit difficult to draw them completely round, but practice makes perfect. If you want a circle that is completely round, you can use a round glass and place it on some paper. Put your pencil on the paper, and while touching the glass, follow the edge of the glass with your pencil until you reach the point where you started.
The circle is useful for many reasons. For example, the circle rolls very well—that’s why a bicycle wheel is circular. Can you find some other things at home that are also circular?
The middle of the circle is called the center. A circle is divided into degrees, ${}^{\circ}$. A whole circle is $360$°. If you cut the circle in half, through the center, you get a semicircle. This part is $180$°.
The distance from the center to the boundary is called the radius. The distance from boundary to boundary through the center is called the diameter.
Rule
A whole circle is $360$°.
Circular sector:
A pizza slice of the circle.
Radius:
The distance from the center to the boundary.
Diameter:
The distance from boundary to boundary through the center.
Now, let’s learn to calculate the circumference and area of a circle. Then you need to know about $\pi $ (pronounced “pie”).
Video Crash Courses
Want to watch animated videos and solve interactive exercises about finding the circumference of a circle?
You can find the circumference of a circle if you wrap a string along the periphery of the circle (its boundary) and measure the length of the string. But to calculate the circumference of a circle, use the formula below. You just need to know the length of the radius or the diameter.
Formula
$$\begin{array}{llll}\hfill C& =2\cdot \pi \cdot r,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill C& =\pi \cdot d,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$ where $r$ is the radius, $d$ is the diameter and $\pi \approx 3.14$.
Example 1
Calculate the circumference of a circle with a radius of $\text{}2\text{}\phantom{\rule{0.17em}{0ex}}\text{cm}$
Use the formula and let $r=2\phantom{\rule{0.17em}{0ex}}\text{cm}$.
$$C=2\cdot \pi \cdot 2\phantom{\rule{0.17em}{0ex}}\text{cm}\approx 12.56\phantom{\rule{0.17em}{0ex}}\text{cm}$$ |
Example 2
Calculate the circumference of a circle with a diameter of $\text{}5\text{}\phantom{\rule{0.17em}{0ex}}\text{cm}$
$$C=\pi \cdot 5\phantom{\rule{0.17em}{0ex}}\text{cm}\approx 15.7\phantom{\rule{0.17em}{0ex}}\text{cm}$$ |
Video Crash Courses
Want to watch animated videos and solve interactive exercises about finding the area of a circle?
Now let’s look at the area of a circle. The unit of area is always raised to the power of 2.
Formula
$$A=\pi \cdot {r}^{2},$$ |
where $r$ is the radius and $\pi \approx 3.14$.
Example 3
Calculate the area of a circle with a radius of $\text{}2\text{}\phantom{\rule{0.17em}{0ex}}\text{cm}$
Use the formula above and insert $r=2\phantom{\rule{0.17em}{0ex}}\text{cm}$:
$$A=\pi \cdot {2}^{2}\approx 3.14\cdot 4=12.56\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}$$ |
Example 4
Calculate the area of a circle with a diameter of $\text{}6\text{}\phantom{\rule{0.17em}{0ex}}\text{cm}$
When calculating the area of a circle, you need the length of the radius to use in the formula. The circle’s diameter is twice as long as the radius, so you can just divide the diameter by 2: $r=\frac{6}{2}=3$. Use the formula above and let $r=3\phantom{\rule{0.17em}{0ex}}\text{cm}$:
$$A=\pi \cdot {3}^{2}\approx 3.14\cdot 9\approx 28.26\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}$$ |
Example 5
Calculate the area of the semicircle with a radius equal to $\text{}5\text{}\phantom{\rule{0.17em}{0ex}}\text{cm}$
The formula calculates the area of the whole circle, but you want the area of the semicircle. That means you divide the whole area by 2:
Math Vault
Want to solve exercises about circles? Try Math Vault!