8.2: Circle Basics, Area, and Perimeter
Learning Objectives
- Distinguish between radius and diameter of a circle and between interior and exterior points.
- Examine inscribed polygons.
- Calculate the circumference of a circle.
- Calculate the area of a circle.
Circle, Center, Radius
A circle is defined as the set of all points that are the same distance away from a specific point called the center of the circle. Note that the circle consists of only the curve but not of the area inside the curve. The distance from the center to the circle is called the radius of the circle.
The ______________________ of a circle is the point in the middle of the circle.
A _______________________ is the set of all points equidistant from the center.
The _______________________ is the distance from the center to the outer edge of the circle.
We often label the center with a capital letter and we refer to the circle by that letter. For example, the circle below is called circle \begin{align*}A\end{align*} or \begin{align*} \bigodot A\end{align*}.
The Parts of a Circle
A circle is the set of all points in a plane that are a given distance from another point called the center. Flat round things, like a bicycle tire, a plate, or a coin remind us of a circle.
Can you think of some other examples of circles in the real world?
The diagram above and the list below review the names for the “parts” of a circle.
- Center: the point in the middle of the circle
- Circle: the points that are a given distance from the center (which does not include the center or interior)
- The interior: all the points (including the center) that are inside the circle
- Circumference: the distance around a circle (exactly the same as perimeter)
- Radius: any segment from the center to a point on the circle (sometimes “radius” is used to mean the length of the segment and it is usually written as \begin{align*}r\end{align*})
- Diameter: any segment from a point on the circle, through the center, to another point on the circle (sometimes “diameter” is used to mean the length of the segment and it is usually written as \begin{align*}d\end{align*})
Reading Check:
Fill in the blanks using the word bank below:
\begin{align*}& \mathbf{diameter} && \mathbf{circle} && \mathbf{circumference}\\ & \mathbf{radius} && \mathbf{interior} && \mathbf{center}\end{align*}
1. The _________________________ of a __________________________ is the point in the middle of it.
2. The ___________________________ is a special word for the perimeter of a circle.
3. The distance from the center to a point on the circle is called the _________________.
4. When you draw a line from one edge of the circle to another, and it goes through the center, it is called a __________________________.
5. The _______________________of a circle is the space inside the entire circle.
A diameter is formed by two collinear radii. This means that when two radius segments fall in the same line, the line is called the diameter. Likewise, a radius is half the length of a diameter:
\begin{align*}d = 2r \qquad \text{or} \qquad r = \frac{d}{2}\end{align*}
The diameter is twice (or two times) the ________________________.
Circumference Formula
The formula for the circumference of a circle is a classic. It has been known, in rough form, for thousands of years. We will look at one way to derive this formula.
Start with a circle with a diameter of 1 unit. Inscribe a regular polygon in the circle.
\begin{align*}\rightarrow\end{align*} Inscribe means to draw the shape inside the circle, where each vertex touches the outer edge of the circle.
We will inscribe regular polygons with more and more sides and see what happens. For each inscribed regular polygon, the perimeter will be given in decimal form (don’t worry about how we got the number):
What do you notice?
- The more sides there are, the closer the polygon is to the circle itself.
- The perimeter of the inscribed polygon increases as the number of sides increases.
- The more sides there are, the closer the perimeter of the polygon is to the circumference of the circle.
You can see that as we add more sides to the polygon inside the circle, it looks like the polygon is closer to “matching up” with the circumference! You will also notice that the perimeter number below each picture is increasing.
Now imagine that we continued inscribing polygons with more and more sides. It would become nearly impossible to tell the polygon from the circle. The table on the next page shows the results if we did this.
Regular Polygons Inscribed in a Circle with a Diameter of 1 unit:
Number of sides of polygon | Perimeter of polygon |
---|---|
3 | 2.598 |
4 | 2.828 |
5 | 2.939 |
6 | 3.000 |
8 | 3.062 |
10 | 3.090 |
20 | 3.129 |
50 | 3.140 |
100 | 3.141 |
500 | 3.141 |
As the number of sides of the inscribed regular polygon increases, the perimeter seems to approach a “limit.” This limit, which is the circumference of the circle, is approximately 3.14159. . . This is the famous and well-known number \begin{align*}\pi\end{align*}.
\begin{align*}\pi\end{align*} is the Greek letter pi (pronounced “pie”) that is often used with circles.
\begin{align*}\pi\end{align*} is an irrational number. This means that it is an endlessly non-repeating decimal number.
We often use \begin{align*}\pi \approx 3.14\end{align*} as a value for \begin{align*}\pi\end{align*} in calculations, but remember that this is only an approximation.
- The number \begin{align*}\pi\end{align*} approximates to ______________________.
Conclusion: The circumference of a circle with diameter 1 is \begin{align*}\pi\end{align*}.
Suppose a circle has a diameter of \begin{align*}d\end{align*} units.
The scale factor of this circle (of diameter \begin{align*}d\end{align*}) and the one with diameter 1, is:
\begin{align*}d : 1, \qquad \frac{d}{1}, \qquad \text{or just} \qquad d\end{align*}.
You already learned how a scale factor affects linear measures, which include perimeter and circumference. Remember: multiply any linear measures by the scale factor. If the scale factor is \begin{align*}d\end{align*}, then the perimeter is \begin{align*}d\end{align*} times as much.
This means that if the circumference of a circle with diameter 1 is \begin{align*}\pi\end{align*}, then we multiply the scale factor, \begin{align*}d\end{align*}, by the circumference:
The circumference of a circle with diameter \begin{align*}d\end{align*} is (\begin{align*}\pi\end{align*} times \begin{align*}d\end{align*}) or \begin{align*}\pi d\end{align*}.
Circumference Formula
Let \begin{align*}d\end{align*} be the diameter of a circle, and \begin{align*}C\end{align*} be the circumference.
\begin{align*}C = \pi d\end{align*}
Since we know that \begin{align*}d = 2r\end{align*}, another way to write the circumference is:
\begin{align*}C = 2 \pi r\end{align*}
Example 1
A circle is inscribed in a square. Each side of the square is 10 cm long. What is the circumference of the circle?
You can see in the picture above that the length of a side of the square is also the diameter of the circle. Use the formula for circumference, \begin{align*}C = \pi d\end{align*} , where \begin{align*}d = 10\end{align*}:
\begin{align*}C &= \pi d\\ C &= \pi (10) = 10 \pi \approx 31.4 \ cm\end{align*}
Note that there are two ways to leave your answer:
- Sometimes we use the decimal approximation \begin{align*}\pi \approx 3.14\end{align*}. In this example, the circumference is 31.4 cm using that approximation.
- An exact answer can be given in terms of \begin{align*}\pi\end{align*} (leaving the symbol \begin{align*}\pi\end{align*} in the answer instead of multiplying it out.) Here, the exact circumference is \begin{align*}10 \pi \ cm\end{align*}.
Reading Check:
1. True/False: The circumference is the measurement of the outside edge (or perimeter) of a circle.
2. True/False: Writing the circumference as \begin{align*}C = \pi d\end{align*} is the same as writing it as \begin{align*}C = 2 \pi r\end{align*} because the diameter is twice the radius.
3. What is the difference between an approximated answer and an exact answer?
(*Hint: In an approximated answer, we use the number ___________ for \begin{align*}\pi\end{align*}, while in an exact answer, we ...)
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Area of a Circle
The area of a circle is a measurement of the space in the interior of the circle.
The area of a circle measures the _______________________ of the circle.
The big idea:
- Find the areas of regular polygons with radius 1.
- Let the polygons have more and more sides.
- See if a limit shows up in the data.
- Use similarity to generalize the results.
The details:
Begin with polygons having 3, 4, and 5 sides, inscribed in a circle with a radius of 1:
Now imagine that we continued inscribing polygons with more and more sides. It would become nearly impossible to tell the polygon from the circle. The table below shows the results if we did this:
Regular Polygons Inscribed in a Circle with a Radius of 1 unit:
Number of sides of polygon | Area of polygon (rounded to the nearest ten thousandth) |
---|---|
3 | 1.2990 |
4 | 2.0000 |
5 | 2.3776 |
6 | 2.5981 |
8 | 2.8284 |
10 | 2.9389 |
20 | 3.0902 |
50 | 3.1333 |
100 | 3.1395 |
500 | 3.1415 |
1000 | 3.1416 |
2000 | 3.1416 |
In looking carefully at the chart on the previous page, you will notice that as the number of sides of the inscribed regular polygon increases, the area seems to approach a “limit.” This limit is approximately 3.1416, which is \begin{align*}\pi\end{align*}.
Conclusion: The area of a circle with radius 1 is \begin{align*}\pi\end{align*}. Therefore, the area of a circle with radius \begin{align*}r\end{align*} is \begin{align*}\pi r^2\end{align*}.
Area of a Circle Formula
Let \begin{align*}r\end{align*} be the radius of a circle, and \begin{align*}A\end{align*} the area.
\begin{align*}A = \pi r^2\end{align*}
You probably noticed that the reasoning about area here is very similar to the reasoning we used earlier in this lesson when we explored the perimeter of polygons and the circumference of circles. What sort of reasoning is this if we based our conclusions on examples? It is inductive reasoning!
Example 2
A circle is inscribed in a square. Each side of the square is 10 cm long. What is the area of the circle?
Like in Example 1, the length of a side of the square is also the diameter of the circle.
Use the formula for area of a circle \begin{align*}A = \pi r^2\end{align*}, where the radius \begin{align*}r\end{align*} is 5 cm:
\begin{align*}A &= \pi r^2\\ A &= \pi(5^2) = 25 \pi \approx 78.5\end{align*}
The area of the circle is exactly \begin{align*}25 \pi \ cm^2\end{align*} or approximately \begin{align*}78.5 \ cm^2\end{align*}.
Reading Check
1. True/False: The area of a circle is the same as the circumference.
2. Why did you answer true or false? Explain.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
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