ACT Math: A Challenging Arc Length & Sector Area Problem

One of our more popular posts involves Arc Lengths & Sector Areas for circles. With that in mind, we thought we’d give you a challenging problem (well, really two problems) involving this topic. Give it a try and then scroll down for the answer.

The circle below has its center at O and triangle AOB is equilateral with sides that are each 6 inches.

1.  What is the area, in square inches, of the shaded region?

A.  6π – 3√3
B.  9π – 3√3
C.  6π – 6√3
D.  6π – 9√3
E.  9π – 9√3

SAT Math: Equations of Circles

The focus of the new SAT has shifted more towards algebra and functions and away from geometry. However, you can still expect to see some questions on topics from geometry, including circles. You should know how to work with the equation of a circle in both its forms.

Standard Form: (x – h)² + (y – k)² = r²
General Form: x² + y² + Cx + Dy + E = 0

In the Standard Form, (hk) is the center of the circle and r is the radius.

The questions that give you the equation in Standard Form should be easier than those that involve General Form. Consider, for example, the following problem.

(Note: Click the image to enlarge)

ACT Math: A Challenging Circle Problem

Here’s another challenging problem for you, this one involving circles. You’ll need to use concepts you’ve learned in both Algebra and Geometry to get this one right. Good luck!

In the circle below with center at Point O, the length of chord AB is 30 units. The radius in the diagram is perpendicular to the chord AB, which cuts the radius into two pieces with lengths of x units and x + 1 units as labeled in the diagram. What is the length, in units, of the radius of the circle?

A.  13
B.  15
C.  17
D.  18
E.  20