Continue working with your circle that has an area of 36π. But now, cut it in half and make it a semicircle. Any time you have a fractional portion of a circle, it’s known as a sector:
What effect does cutting the circle in half have on the basic elements of the circle? The diameter stays the same, as does the radius. But what happened to the area and the circumference? They’re also cut in half. So the area of the semicircle is 18π and the remaining circumference is 6π. When dealing with sectors, the portion of the circumference that remains is called the arc length. So the arc length of this sector is 6π.
In fact, this rule applies even more generally to circles. If, instead of cutting the circle in half, you had cut it into quarters, each piece of the circle would have 1/4 the area of the entire circle and 1/4 the circumference:
Now, on the GRE, you’re unlikely to be told that you have one-quarter of a circle. There is one more basic element of circles that becomes relevant when you are dealing with sectors, and that is the central angle. The central angle of a sector is the degree measure between the two radii. Take a look at the quarter circle. There are 360° in a full circle. What is the degree measure of the angle between the two radii? The same thing that happens to area and circumference happens to the central angle. It is now 1/4 of 360°, which is 90°:
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How can you use the central angle to determine sector area and arc length? For the next example, you will still use the circle with area 36π, but now the sector will have a central angle of 60°:
You need to figure out what fractional amount of the circle remains if the central angle is 60°. If 360° is the whole amount, and 60° is the part, then 60/360 is the fraction you’re looking for, and 60/360 reduces to 1/6. That means a sector with a central angle of 60° is 1/6 of the entire circle. If that’s the case, then the sector area is
and arc length is
. So:
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In this last example, you used the central angle to find what fractional amount of the circle the sector was. But any of the three properties of a sector (central angle, arc length, and area) could be used if you know the radius.
Here’s an example:
A sector has a radius of 9 and an area of 27π. What is the central angle of the sector?
You still need to determine what fractional amount of the circle the sector is. This time, however, you have to use the area to figure that out. You know the area of the sector, so if you can figure out the area of the whole circle, you can figure out what fractional amount the sector is.
You know the radius is 9, so you can calculate the area of the whole circle. Area = πr2, so Area = π(9)2 = 81π. Because
, the sector is 1/3 of the circle. The full circle has a central angle of 360°, so the central angle of the sector is 1/3 × 360 = 120°:
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Now recap what you know about sectors. Every question about sectors involves determining what fraction of the circle the sector is. That means that every question about sectors will provide you with enough information to calculate one of the following fractions:
Once you know any of those fractions, you know them all, and, if you know any specific value, you can find the value of any piece of the sector or the original circle.
A sector has a central angle of 270° and a radius of 2. What is the area of the sector?
A sector has an arc length of 4π and a radius of 3. What is the central angle of the sector?
A sector has an area of 40π and a radius of 10. What is the arc length of the sector?