ACT Math: Arc Lengths and Sector Areas

As you’re reviewing for the geometry problems that appear on the ACT Math test, one of the things you’ll want to go over is circles. There are many types of problems that can appear on the test involving circles. This post will review two of those: arc length and sector area.

You can find both arc length and sector area using formulas. Or you can take a more “common sense” approach using what you know about circumference and area. Let’s look at both of these concepts using the following problems.

A circle with its center at Point O has a radius of 12 cm. If ∠AOB has a measure of 60 degrees, what is the length of arc AB?

Arc Length

A.  2π
B.  4π
C.  6π
D.  12π
E.  24π

Method 1: Use the arc length formula
There is a formula, sometimes taught in geometry but often not until pre-calculus, that computes the arc length of a circle. If you know the circle’s radius and the measure of the central angle (in radians), then the arc length can be found using s = θ.

The first thing you’ll need to do for this problem is convert the 60 degree angle to radians. This can be done by mulitplying it by π/180. In radians our angle is 60(π/180) = 60π/180 = π/3. Now use the arc length formula to determine the length of arc AB: s = θ = 12(π/3) = 4π. The correct answer is Choice B.

Method 2: Use what you know about circumference
But what if you’ve never seen this formula? Or what if you haven’t taken pre-calculus yet and you’re not quite sure what a radian is? You can still get the problem correct using what you know about circumference! Let’s see how. You know the circumference of this circle is C = 2πr = 2π(12) = 24π. The arc length we are looking for is some part, or some fraction, of this total circumference. But what fraction? Recall that a full circle is 360 degrees and that our central angle is 60 degrees, so it stands to reason that the arc length is 60/360, or 1/6, of the total circumference. That would make the arc length (1/6)(24π) = 4π …… same answer we got using the formula!

The circle below has its center at Point P. What is the area of the shaded region if the measure of ∠XPY is 45 degrees and the radius of the circle is 8?

Sector Area

F.  4π
G.  8π
H.  16π
J.  32π
K.  64π

Method 1: Use the sector area formula
Once again, there is a formula you can use to find the sector area:  A = ½r²θ. Just as in the arc length formula, you need to know the radius of the circle and the central angle.

Begin by converting our central angle of 45 degrees to radians: (45)(π/180) = 45π/180 = π/4. Now apply the formula to get the area of the sector (the shaded region): A = ½r²θ  = ½(8²)(π/4) = ½(64)(π/4) = 8π. The correct answer is Choice G.

Method 2: Use what you know about area
First let’s compute the area of the circle: A = πr² = π(8²) = 64π. The shaded region is some fraction of the circle, and we can compute that fraction just like we did in the arc length problem above. Our central angle is 45 degrees and the entire circle is 360 degrees, so the shaded region is 45/360, or 1/8, the area of the entire circle. That makes the area of the shaded region equal to (1/8)(64π) = 8π. Once again, the “common sense” approach got us the same answer as the fancy pre-calculus formula.

The moral of the story is this: On many ACT Math problems, there is more than one approach that will get you the correct answer. If you don’t remember a formula, see if you can logic your way through the problem and arrive at the correct answer in a different way!

If you’d like to try a couple of challenging problems involving arc length and sector area, head on over here.

If you have questions about these problems or anything else to do with the ACT, leave a comment below or drop us an email at info@cardinalec.com.

 

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