GMAT Geometry: Central Angles
In poetry, a rose is a rose is a rose. On GMAT problems with central angle “slices” in circles, a fraction is a fraction is a fraction.
This may seem like common sense. Cut a pizza into six slices. If you cut it evenly, each slice now has one-sixth the cheese, one sixth the crust, and an angle of one sixth the way around a circle—that is, 60 degrees. However, though this may seem obvious, it’s actually a very useful technique for resolving certain geometry problems.
Consider the following Data Sufficiency question:
In a radius 6 circle, two points A and B are connected to the center, point O. What is angle AOB?
1) The length of the minor arc defined by sector O is 1.5π
2) The area of the sector defined by angle AOB is 4.5π
It looks all GMAT-like and formal, but if you actually think about it…, it’s just the pizza I described above. We’re taking a slice, and want to know what’s the angle of that slice. The “cheese” of the pizza is the area of a radius 6 circle, which is the radius squared times pi, or 36π. The “crust” is it’s circumference, which in this case would be the diameter times pi, 12π.
Statement 1) tells us that the length of the arc/crust is 1.5π, which is 1.5/12 = 1/8 of the circumference. And an eighth is an eighth is an eighth. Our slice, which goes one eighth of the way around the outside of the circle, and it goes an eighth of the way around the inside of our circle as well—its central angle is 1/8 * 360 = 45 degrees. Sufficient!
And statement 2) says that the area of our slice/sector is 4.5π, or 4.5/36 = 1/8 the total area of the whole pie. Once again, an eighth is an eighth is an eighth, so by the exact same reasoning we get 45 degrees. Sufficient!
This is a pretty basic rule, but it’s widely applicable to many circle problems. It can be especially useful with central triangles, which have central angles by definition. Take a look at today’s question of the day to test your skills, and when you get it right, treat yourself to a slice of pepperoni as a reward!
What is the area of the circle above with center O?
(1) The area of D AOC is 18.
(2) The length of arc ABC is 3π.
Step 1: Analyze the Question Stem
This is a Value question. For any Circle question, we
only need one defining parameter of a circle (area,
circumference, diameter, or radius) in order to calculate
any of the other parameters. Also, all radii of the same
circle will have the same length. So AO = CO. That makes
the triangle an isosceles right triangle (or 45-45-90) for
which we know the ratio of the sides. Not only would the
circle’s circumference, diameter, or radius be sufficient,
but information that gave us any side length of triangle
AOC would also be sufficient, as it would give us the length
of the radius.
Step 2: Evaluate the Statements
Statement (1): We are given the area, and we already know
that the base and the height are equal. So if we call the
radius of the circle r, then the area of the triangle is equal to
1/2 (Base)(Height) = 1/2 (r)(r) = 1/2 r ^2 = 18
Remember that we are not asked to calculate the actual
value of r. Because we have set up an equation with
one variable, and we know in this case that r can only
be positive since it is part of a geometry figure, we have
enough information to determine the area of the circle (the
solution would have been r ^2 = 36, r = 6).
Therefore, Statement (1) is sufficient. We can eliminate
choices (B), (C), and (E).
Statement (2): Because O is the center of the circle and
angle AOC measures 90 degrees, we know that the length
of arc ABC is one-fourth of the circumference. Because this
statement allows us to solve for the circumference, it is
(D) is correct.