In order to work on the final section of our study of areas, we must first learn about a shape that we have not discussed at all in the past. In fact, this shape is not a polygon at all, which means that it doesn't have vertices or sides. The shape we are going to focus on in this section is called a circle. Let's begin this lesson by examining the definition of a circle.
Definition: A circle is a plane curve that is equidistant from a given fixed point, called the center.
Every point on the curve is a distance of x units away from center C.
There are countless properties we can use in order to describe circles, but we will just focus on two of them: radius and diameter. Although we do not directly use diameter to find the area of a circle, understanding how it compares to the radius can help us figure out areas of circles. Let's look at the definitions of radius and diameter, as well as the illustration below to see how they relate.
Definition: A radius is a line segment that joins the center of a circle with any point on the circumference of the circle.
Definition: A diameter is a straight line segment that passes through the center of a circle.
Notice that the radius is only extended from the center of the circle to the outside edge of the circle, whereas the diameter goes all the way through from one side to the other. Because the definition of a circle describes the locus of points that are equidistant from the center, we know that all of the radians of a circle are equal. So, we can essentially just break down the diameter at the center of the circle to create two radians. Therefore, we know that
where d is the length of the diameter of a circle and r is the length of the radius.
Because all of the radians of a circle are equal, we know that two of them make up the diameter of a circle.
Now that we've discussed the important parts of a circle, we can learn how to measure the areas of circles. Because circles don't have sharp edges, their areas in square units will almost never come out to an even number, so we will just round areas to the hundredths place. Let's learn how to use the area formula for circles.
The area of a circle is equal to the product of pi (?), and the radius of the circle squared. We describe this mathematically as
where A is the area in square units, ? is a mathematical constant (approximately equal to 3.14), and r is the radius.
Recall that we were able to put the diameter of a circle in terms of its radius. Because every problem will not give us the radius of a circle, we might need to use our knowledge of their diameters to help us figure out their areas. In other words, if we are given the diameter of a circle, we know that half of the diameter is equal to the radius, which we can plug into our area formula. Let's work on some exercises now.
Find the area of circle p.
Let's analyze the information we've been given in order figure out what we must do to find the area of circle p. We are given that the diameter is 18 inches, and we know that the diameter of a circle is twice and long as its radius, so we all we have to do in order to find the radius is take half of the diameter.
We see that the radius of our circle is 9 inches.
Remember, that ? is not a variable; it is a mathematical constant. Also, we will not worry about much precision when it comes to the value of ? . We can just define ? as 3.14 since our final answer will be rounded to the hundredths place. We are ready to solve for the area of circle p.
So, the area of circle p is approximately 254.34 square inches.
Now, let's look at another example that will require a bit more work.
Find the value of x if the area of circle g is approximately 1661 square feet.
In this example, we have been given the area of circle g, so we will have to work backwards in order to find the radius. Let's fill out the area formula, substituting in for the variables we know.
While ? is not equal to 3.14, we will use 3.14 as an approximate value to help us solve for x. We have
In order to get rid of the square, we need to take the square root of both sides:
We know that the radius of circle g is approximately 23 feet, but we haven't solved for the variable x. We just need to subtract 7 from both sides of the equation, and we get
So, the value of x is approximately 16.
Let's learn about sectors of circles now.
Sometimes, we will not want to find the areas of full circles and instead need to find smaller sections of a circle. In these cases, we will need a way to calculate these parts of circles called sectors. Let's study the definition of sectors and see what they look like before we introduce the area formula.
Definition: A circular sector is the portion of a circle enclosed by two radii and an arc of the circle.
Notice that the arc of a circle is just the part of the circumference enclosed by the endpoints of both radians.
Working with the sectors of circles can be quite simple if we know how to apply the area formula for circles. If we know that the circle is split up into a certain amount of congruent areas, we can just put the corresponding factor into our area formula. For instance, if we have a circle that is split up into four equal sections, and we want to find the area of one of those sections, our area formula would be
Because one-fourth of the circle is shaded, we just multiply the area formula of circle c by a factor of ¼ to find the area of the circle sector.
In other cases, we may be given the measure of the angle at the radius of a circle, called the central angle. For those exercises, we can apply the area of sectors formula, which is
where A represents area, x represents the degree measure of the central angle, and r is the radius.
This formula essentially does the same as what we've done in the previous example because it just converts the degree measure of the interior angle into an equivalent fraction. Circles have degree measures of 360°. Therefore, when we divide a given measure by 360°, we are just taking the fraction of the circle we desire and multiplying it by our regular area formula. Let's take a look at one final example to make sure we understand how to apply the area of sectors formula.
Find the area of the shaded sector below.
Because circle t has not been split into even sections for us, we cannot just multiply the area formula of circles by a fraction. Rather, we need to use the degree measure of the central angle and plug it into the area of sectors formula. Remember, we need to also use the fact that the radius is 14 meters long in order to solve for the area of the sector. Let's do this now.
So, the area of the shaded sector is approximately 230.79 square meters.
Now, let's go back and analyze the relationship between the degree measure of the central angle and the portion of the circle that was shaded.
The first factor of the area formula for sectors eventually simplified to ? because
The fact that this fraction simplifies to ? means that the sector's area is three-eighths the area of the entire circle.
If we split up circle t into eight congruent pieces, we see that a 135° central angle creates a sector with three-eighths the area of the entire circle.
Now, we know how to measure smaller sections of a circle and can compare these sections to the area of the circle as a whole.
While we have yet to discuss circles in depth, working with sectors of circles can help us learn how to relate the degrees and radians of circles, which are significant components of subjects like precalculus and calculus.