8.2: Area of Circles
Introduction
The Discus Ring
“I don’t know how to figure this out,” Jesse said to his friend Emory one morning.
“Figure what out?” Emory inquired.
“I have to figure out the area of the discus ring. That is what Mrs. Henry asked me to figure out,” Jesse said.
“Well, what do you know?”
“I know that the shape of it is a circle. I also know that the diameter of the circle is 8 feet. I need the area of the ring now and that is where I am stuck,” Jesse explained.
“That’s not so hard,” Emory said.
Jesse looked at his friend puzzled.
Do you know what Emory knows? In this lesson you will learn all about area and circles. At the end of the lesson, you will see this problem again. Then you will need to help Jesse solve for the area of the discus ring.
What You Will Learn
By the end of this lesson, you will understand how to perform the following skills.
- Identify the radius, diameter and circumference of circles.
- Model the area of a circle as the sum of the areas of several congruent sectors reformed to approximate a parallelogram.
- Find areas and linear dimensions of circles and sectors of circles.
- Solve real – world problems involving areas of circles and sectors of circles, including metric and customary units of length and area.
Teaching Time
I. Identify the Radius, Diameter and Circumference of Circles
Circles are unique geometric figures. A circle is the set of points that are equidistant from a center point.
The radius of a circle is the distance from the center to any point on the circle. The diameter is the distance across the circle through the center. The diameter is always twice as long as the radius.
We also use the special number pi when dealing with circle calculations. Pi is a decimal that is infinitely long (3.14159265...), but in our calculations we round it to 3.14. We use the symbol \begin{align*}\pi\end{align*} to represent this number.
Pi is the ratio of the circumference, or distance around a circle, to the diameter. In other words, these two measurements are related. If we change the diameter, the circumference changes proportionally. For example, if we double the length of the diameter, the circumference doubles also.
Let’s review working with the radius and diameter while finding the circumference.
As the diameter of the circle grows, the circumference of the circle grows at the same rate. In other words, however the diameter of the circle changes, the circumference of the circle must change exactly the same way. This is a proportional relationship.
We express this proportional relationship as a ratio. A ratio simply means that two numbers are related to each other. Circles are special in geometry because this ratio of the circumference and the diameter always stays the same.
We can see this when we divide the circumference of a circle by its diameter. No matter how big or small the circle is, we will always get the same number. Let’s try it out on the circles below.
\begin{align*}\frac{Circumference}{Diameter} &=\frac{6.28}{2}=3.14\\ \frac{Circumference}{Diameter} &=\frac{12.56}{4}=3.14\end{align*}
Even though we have two different circles, the result is the same. Therefore the circumference and the diameter always exist in equal proportion, or a ratio, with each other. Whenever we divide the circumference by the diameter, we will always get 3.14, pi.
Using the equations above, we can write a general formula that shows the relationship between pi, circumference, and diameter. When we rearrange it, we get the formula for the circumference of a circle.
\begin{align*} \pi=\frac{C}{d}\end{align*} so \begin{align*}C=\pi d\end{align*}
If we divide the circumference by the diameter to find pi, then we can use the formula circumference equals pi times the diameter to find the circumference of any circle.
Example
What is the circumference of a circle that has a diameter of 3 inches?
To find the circumference, we can substitute these values into the formula.
\begin{align*}C &=\pi(3)\\ C &=3.14(3)\\ C &=9.42 \ inches \end{align*}
What about if we were given the measurement for the radius instead of the diameter?
Well, we know that the radius is one – half of the diameter, so we can use the following formula or you can figure out the measurement for the diameter by using mental math.
\begin{align*}C=2 \pi r\end{align*}
Example
What is the circumference of a circle if the radius is 2.5 feet?
First, we can find the diameter using this measurement. If the radius is 2.5 feet, then the diameter is 5 feet. Let’s find the circumference using this measurement.
\begin{align*} C &=3.14(5)\\ C &=15.7 \ feet\end{align*}
We could also have used the radius alone to find the circumference. We just use a different formula.
\begin{align*}C &=2(3.14)(2.5)\\ C &=15.7 \ feet\end{align*}
You can see that we can use either the measurement for the radius or for the diameter to find the measurement for the circumference.
Write both of these formulas down in your notebook.
II. Model the Area of a Circle as the Sum of the Areas of Several Congruent Sectors Reformed to Approximate a Parallelogram
Area is the amount of two-dimensional space a figure takes up. In other words, area is the space contained within a circle’s circumference.
In rectangles, we know that area is a measure of the length times the width (the two dimensions). Circles are curved, so how can we measure its length and width?
Well, we can cut up a circle into smaller portions, called sectors. A sector is a part of a circle with radii for two sides and part of the curved circumference as another. Sectors look like pie slices.
We can arrange the sectors of a circle to approximate a rectangle. Take a look at this picture.
To find the area of the rectangle, we multiply the two dimensions, length and width. This gives us the formula \begin{align*}A = lw\end{align*}. We can do the same for the sectors that have been arranged to form a rectangle. This gives us \begin{align*}A =\pi r \times r\end{align*}, or \begin{align*}\pi r^2\end{align*}. Therefore the formula for finding the area of circles is here.
\begin{align*}A=\pi r^2\end{align*}
We already know that the symbol \begin{align*}\pi\end{align*} represents the number 3.14, so all we need to know to find the area of a circle is its radius. We simply put this number into the formula in place of \begin{align*}r\end{align*} and solve for the area, \begin{align*}A\end{align*}.
We can use this formula whether we have been given the radius or the diameter of the circle.
III. Find Areas and Linear Dimensions of Circles and Sectors of Circles
Now that you have the formula for finding the area of a circle, we can apply it when working with examples. Let’s try out the formula.
Example
What is the area of the circle below?
We know that the radius of the circle is 12 centimeters. We put this number into the formula and solve for \begin{align*}A\end{align*}.
\begin{align*}A & =\pi r^2\\ A &=\pi (12^2)\\ A &= 144 \pi\\ A & = 452.16 \ {cm^2}\end{align*}
Remember that squaring a number is the same as multiplying it by itself. The area of a circle with a radius of 12 centimeters is 452.16 square centimeters when we approximate pi as 3.14. We always show area in square units. Let’s try another.
Example
What is the area of a circle with a diameter of 45 centimeters?
Read the problem carefully! We need to find the area, but what information is given in the problem? This time we know the diameter, not the radius. How can we find the radius so that we can use the area formula?
We know that the diameter of a circle is always twice the length of the radius. If the diameter is 45 centimeters, then the radius must be \begin{align*}45 \div 2 = 22.5 \ centimeters\end{align*}. Now we can put this number into the formula.
\begin{align*}A &=\pi r^2\\ A &=\pi(22.{5^2})\\ A &= 506.25 \pi\\ A &= 1,589.63 \ {cm^2}\end{align*}
The area of a circle with a diameter of 45 centimeters (and a radius of 22.5 centimeters) is 1,589.63 square centimeters when we approximate pi as 3.14.
Nice work! We can also use the formula to find the radius or diameter if we know the area. Let’s see how this works.
Example
The area of a circle is 113.04 square inches. What is its radius?
This time we know the area and we need to find the radius. We can put the value for area into the formula and use it to solve for the radius, \begin{align*}r\end{align*}.
\begin{align*}A &=\pi r^2\\ 113.04 &= \pi r^2\\ 113.04 \div \pi & = r^2\\ 36 & = r^2\\ \sqrt{36} &= r\\ 6 \ in.& = r\end{align*}
To solve this problem, we need to isolate the variable \begin{align*}r\end{align*}. First we divide both sides by \begin{align*}\pi\end{align*}, or 3.14. Then, to remove the exponent, we take the square root of both sides. A square root is a number that, when multiplied by itself, gives the number shown. We know that 6 is the square root of 36 because \begin{align*}6 \times 6 = 36\end{align*}.
The radius of a circle with an area of 113.04 square inches is 6 inches.
Using what we have learned, can we find the area of a sector?
Sometimes we may be asked to find the area of a sector, or portion, of a circle, such as a quarter or half of the circle. As long as we know the radius, we can find the area of the whole circle. Then we can divide that area into smaller pieces or subtract a portion to find the area of part of the circle. Let’s try this out.
Example
What is the area of the figure below?
This figure is a quarter of a circle, formed by a \begin{align*}90^\circ\end{align*} angle. Remember, circles contain \begin{align*}360^\circ\end{align*}. One-quarter of \begin{align*}360^\circ\end{align*} is \begin{align*}90^\circ\end{align*}. We know that the radius of the whole circle is 8.5 inches because the two sides of the sector are radii of the circle. Let’s use this value to solve for the area of the whole circle first.
\begin{align*}A & =\pi r^2\\ A &= \pi (8.5^2)\\ A &= 72.25 \pi\\ A &= 226.87 \ in.^2\end{align*}
We know that the area of a whole circle with a radius of 8.5 inches is 226.87 square inches.
Therefore the quarter circle formed by the \begin{align*}90^\circ\end{align*} angle must have \begin{align*}\frac{1}{4}\end{align*} of this area. We can divide the area by 4 to find the area of the sector: \begin{align*}226.87 \div 4 = 56.72\end{align*} square inches. As long as we can find the area of a whole circle, we can divide or subtract to find the area of a sector of a circle.
Write down how you can find the area of a circle and the area of a sector in your notebook.
We can use this formula to find the area of a sector when we know the measure of the angle too. Let’s think about how we can do this in the following example.
Example
What is the area of the sector below?
We know that the angle of the sector is \begin{align*}45^\circ\end{align*} and that the total number of degrees in a circle is always \begin{align*}360^\circ\end{align*}. We can use these to find the fraction of the circle’s area that the sector makes up.
\begin{align*}A &=\frac{degrees \ in \ angle}{degrees \ in \ circle} \times \pi {r^2}\\ A & =\frac{45}{360} \pi {(5^2)}\\ A &=\frac{1}{8} (25) \pi \\ A & =\frac{25}{8} \pi \\ A &=3.125 \pi \\ A &=9.81 \ {cm^2}\end{align*}
The area of this sector is 9.81 square centimeters.
The sector makes up exactly \begin{align*}\frac{1}{8}\end{align*} of the circle, so we know that 9.81 must be \begin{align*}\frac{1}{8}\end{align*} of the circle’s total area. Let’s check to make sure by finding the area of the circle.
\begin{align*}A &=\pi {r^2}\\ A & =\pi {(5^2)}\\ A &= 25 \pi \\ A &= 78.5 \ {cm^2}\end{align*}
Our work is accurate and correct.
IV. Solve Real – World Problems Involving Areas of Circles and Sectors of Circles, Including Metric and Customary Units of Length and Area
We can use the formula we have learned to solve real-world problems involving the area of circles. First, be sure you understand what the question is asking. Do you need to find the area of a circle or a sector, or the radius or diameter? Second, make sure you know what the radius of the circle is. If you have been given the diameter, divide it in half to find the radius.
Let’s start with an example.
Example
Some students have formed a circle to play dodge ball. The radius of the circle is 21 feet. What is the area of their dodge ball circle?
The dodge ball court forms a circle, so we can use the formula to find its area. We know that the radius of the circle is 21 feet, so let’s put this into the formula and solve for area, \begin{align*}A\end{align*}.
\begin{align*}A &=\pi {r^2}\\ A &=\pi {(21)^2}\\ A &= 441 \pi \\ A &= 1,384.74 \ ft^2\end{align*}
Notice that a circle with a large radius of 21 feet has a large area: 1,384.74 square feet.
Now let’s use what we have learned to solve the problem from the introduction.
Real-Life Example Completed
The Discus Ring
Here is the original problem once again. Reread it and then solve for the area of the discus ring.
“I don’t know how to figure this out,” Jesse said to his friend Emory one morning.
“Figure what out?” Emory inquired.
“I have to figure out the area of the discus ring. That is what Mrs. Henry asked me to figure out,” Jesse said.
“Well, what do you know?”
“I know that the shape of it is a circle. I also know that the diameter of the circle is 8 feet. I need the area of the ring now and that is where I am stuck,” Jesse explained.
“That’s not so hard,” Emory said.
Jesse looked at his friend puzzled.
Now solve for the area of the discus ring.
Solution to Real – Life Example
To solve this problem, let’s begin by looking at the known information. We know that the circle is the shape of the discus ring. We also know the diameter of the ring is 8 feet. This information is all that we need.
Let’s look at the formula for finding the area of a circle.
\begin{align*}A=\pi {r^2}\end{align*}
We know that the diameter of the circle is 8 feet. The radius is unknown. Radius is \begin{align*}\frac{1}{2}\end{align*} of the diameter so the radius of the discus ring is 4 feet.
Now we can substitute the given information into the formula and solve.
\begin{align*}A &= \pi {r^2}\\ A &= (3.14)(4^2)\\ A &= (3.14)(16)\\ A &= 50.24 \ sq. feet\end{align*}
This is the area of the discus ring.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Circle
- all points are equidistant from a center point.
- Radius
- the distance half-way across a circle.
- Diameter
- the distance across a circle.
- Circumference
- the distance around a circle.
- Area
- the measurement of the two – dimensional space inside a circle.
- Sector
- the measurement of a section of a circle.
Time to Practice
Directions: Find the circumference of each circle given the radius or diameter.
- \begin{align*}d=10 \ in\end{align*}
- \begin{align*}d=5 \ in\end{align*}
- \begin{align*}d=7 \ ft\end{align*}
- \begin{align*}d=12 \ mm\end{align*}
- \begin{align*}d=14 \ cm\end{align*}
- \begin{align*}r=4 \ in\end{align*}
- \begin{align*}r=6 \ meters\end{align*}
- \begin{align*}r=8 \ ft.\end{align*}
- \begin{align*}r=11 \ in\end{align*}
- \begin{align*}r=15 \ cm\end{align*}
Directions: Find the area of each circle given the radius.
- \begin{align*}r=4 \ in\end{align*}
- \begin{align*}r=3 \ ft\end{align*}
- \begin{align*}r=2.5 \ in\end{align*}
- \begin{align*}r=5 \ cm\end{align*}
- \begin{align*}r=3.5 \ in\end{align*}
- \begin{align*}r=9 \ mm\end{align*}
- \begin{align*}r=11 \ cm\end{align*}
- \begin{align*}r=10 \ in\end{align*}
- \begin{align*}r=7 \ ft\end{align*}
- \begin{align*}r=8 \ in \end{align*}
Directions: Find the area of each sector given the radius and the angle measure. You may round to the nearest hundredth as needed.
- \begin{align*}45^\circ\end{align*} angle with a radius of 3 in.
- \begin{align*}55^\circ\end{align*} angle with a radius of 4 mm
- \begin{align*}60^\circ\end{align*} angle with a radius of 5 cm
- \begin{align*}43^\circ\end{align*} angle with a radius of 6 in
- \begin{align*}70^\circ\end{align*} angle with a radius of 2 in.