Have you ever thrown a discus? Take a look at this dilemma.
“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.
Here you will learn all about area and circles. At the end of the Concept, you will know how to find the area of the discus ring.
Guidance
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.
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.
Take a look.
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.
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.
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.
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.
Find the area of each circle by using the given dimension.
Example A
Diameter = 8 inches
Solution: \begin{align*}50.24 \ sq. in.\end{align*}
Example B
Radius = 3 feet
Solution: \begin{align*}28.26 \ sq. ft.\end{align*}
Example C
Radius = 2.5 inches
Solution: \begin{align*}19.63 \ sq. in.\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
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
- 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.
Guided Practice
Here is one for you to try on your own.
What is the area of a circle with a diameter of 45 centimeters?
Solution
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.
Video Review
Practice
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.