7.5 SAT Math - Circles

Basic facts to know:
The number of degrees in a circle is 360°.
The area of a circle is πr².
The circumference of a circle is 2πr or πd.
-SAT Math Tip: Don't get the formulas for area and circumference confused!  Many students do, and so the SAT testmakers almost always have the other one as an option in the answer choices.
SAT Math Hint: 'circumference' is the same thing as 'perimeter' or distance around.  Occasionally a question will ask for perimeter of a circle; you need to use the circumference formula.  There is no "perimeter" formula for circles.

Radius

A radius extends from the center of a circle to the edge.  As a result, all radii (plural of radius) in a circle are equal.

Diameter

A diameter extends from one end of the circle to the other, passing through the center.  A chord extends from one end of the circle to the other, but generally does NOT pass through the center.  As a result, the diameter is the longest chord in a circle.

Tangent

A tangent is a line that intersects a circle in exactly one point.  Lines tangent to the circle usually touch the circle on the outside, like this:

Students Starting SAT Score 650 and above:

The angle where a radius meets a tangent is always a 90° angle.  Also, if a radius intersects a chord at a 90° angle, the chord is bisected:

One of the more helpful circle rules to know on this test is the relationship between arcs and sectors.  This shows up quite often on the SATs.

Arc

An arc is a length of the circle's edge.  As a result, it is also a small portion, or a certain fraction, of the circle's total circumference. 

Sector

A sector is a slice of the circle, as if it were a pie.  A sector is a fraction of the circle's total area. 

The key to solving these kinds of problems is to think about the circle like a pie.  If you cut a large piece of pie, say 1/4 of the entire pie, you're getting 1/4 of the filling, 1/4 of the crust, etc.  But knowing that circles are measure in degrees, you could figure out mathematically what degree angle you cut in the center of that pie to take that piece – in this case, 1/4 of 360, which is 90°.  And if you imagine a pie in your head, and you've taken 1/4 of it, yeah, 90° looks about right.

The fraction to keep in mind when dealing with arcs and sectors, then, is

This is the essential part of the larger formula that relates arcs and sectors to that fraction:

Try it on the following problem:

14. The points J, K, and L lie on circle O that has a radius of 3.  If angle JOK is 40°, what is the length of arc JOK?

(A) π/3
(B) 2π/3
(C) π
(D) 2π
(E) 4π/3

If you remember the First Geometry Rule, Draw It, you should draw on your paper a circle with center O, and draw an angle JOK that looks about 40°.  Having the visual in front of you will help you solve.  From there, set up your ratios:

Since you know your radius, you can fill in that information and solve:

The answer is B).

In arcs/sectors problems, you won't always be given the angle.  Sometimes you'll be given the ratio of arc to arc length and have to solve for the angle; sometimes you'll be given the wider angle of the circle and you'll need to solve for both the smaller angle they're asking about as well as the ratio they want; sometimes you'll be given the sector area and asked to solve for the arc length…  No matter what they give you, set up the fraction using the information you know and solve for the information you need.  That's it.