Sectors, Areas, and Arcs

If we start with a circle with a marked radius line, and turn the circle a bit, the area marked off looks something like a wedge of pie or a slice of pizza; this is called a "sector" of the circle, and the sector looks like the green portion of this picture:

The angle marked off by the original and final locations of the radius line (that is, the angle at the center of the pie / pizza) is the "subtended" angle of the sector. This angle can also be referred to as the "central" angle of the sector. In the picture above, the central angle is labelled as "θ" (which is pronounced as "THAY-tuh").

What is the area A of the sector subtended by the marked central angle θ? What is the length s of the arc, being the portion of the circumference subtended by this angle?

To determine these values, let's first take a closer look at the area and circumference formulas. The area and circumference are for the entire circle, one full revolution of the radius line. The subtended angle for "one full revolution" is 2π. So the formulas for the area and circumference of the whole circle can be restated as:

What is the point of splitting the angle value of "once around" the circle? I did this in order to highlight how the angle for the whole circle (being 2π) fits into the formulas for the whole circle. This then allows us to see exactly how and where the subtended angle θ of a sector will fit into the sector formulas. Now we can replace the "once around" angle (that is, the 2π) for an entire circle with the measure of a sector's subtended angle θ, and this will give us the formulas for the area and arc length of that sector:

Confession: A big part of the reason that I've explained the relationship between the circle formulas and the sector formulas is that I could never keep track of the sector-area and arc-length formulas; I was always forgetting them or messing them up. But I could always remember the formulas for the area and circumference of an entire circle. So I learned (the hard way) that, by keeping the above relationship in mind, noting where the angles go in the whole-circle formulas, it is possible always to keep things straight.

• Given a circle with radius r = 8 units and a sector with subtended angle measuring 45°, find the area of the sector and the length of the arc.

Notice how I put "units" on my answers. If they'd stated a specific unit for the radius, like "centimeters" or "miles" or whatever, then I could have been more specific in my answer. As it was, I had to be generic.

Many times, if the question doesn't state a unit, or just says "units", then you can probably get away without putting "units" on your answer. However, this often leads to the bad habit of ignoring units entirely, and then — surprise! — the instructor counts off on the test because you didn't include any units. It's probably better to err on the side of caution, and always put some unit, even if it's just "units", on your answers.

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• Given a sector with radius r = 3 cm and a corresponding arc length of 5π radians, find the area of the sector.

Sometimes, an exercise will give you information, but, like the above, it might not seem like it's the information that you actually need. Don't be afraid to fiddle with the values and the formulas; try to see if you can figure out a back door in to a solution, or some other manipulation that'll give you want you need. It's okay not to know, right at the beginning, how you're going to reach the end. (And, if they give you, or ask for, the diameter, remember that the radius is half of the diameter, and the diameter is twice the radius.)

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• A circle's sector has an area of 108 cm², and the sector intercepts an arc with length 12 cm. Find the diameter of the circle.