Cracking The SAT Premium

Note that these relationships are all proportions. Arcs and sectors are proportional to the circumference
and area, respectively, as the central angle is to 360°.

Questions on the Math Test sometimes refer to “minor” or “major” arcs or sectors. A minor arc or sector
is one that has a central angle of less than 180°, whereas a major arc or sector has a central angle greater
than 180° (in other words, it goes the long way around the circle). Let’s see how arcs and sectors might
show up in a problem.

15.Points   A   and B   lie on  circle  O   (not    shown). AO  =   3   and  AOB    =   120°.   What    is  the area

of  minor   sector  AOB?

A)

B) π

C) 3π

D) 9π

Here’s How to Crack It

Because O is the name of the circle, it’s also the center of the circle, so AO is the radius. AOB is the
central angle of sector AOB, so you have all the pieces you need to find the sector. Put them into a
proportion:

Cross-multiply to get 360x = 1,080π (remember to not multiply out π). Divide both sides by 360 and you
get x = 3π, which is (C).

Relationship Between Arc and Angle in Radians

Sometimes you’ll be asked for an arc length, but you’ll be given the angle in radians instead of degrees.
Fear not! Rather than making the problem more complicated, the test writers have actually given you a
gift! All you need to do is memorize this formula:

s   =   rθ

In this formula, s is the arc length, r is the radius, and θ is the central angle in radians. If you know this
formula, these questions will be a snap!

Rectangles and Squares