**Circles**

A circle is a set of points in a plane equidistant from a fixed point (the center of the circle). The perimeter of a circle is called the *circumference.*

A line segment from a circle to its center is a *radius.*

A line segment with both end points on a circle is a *chord.*

A chord passing through the center of a circle is a *diameter.*

A diameter can be viewed as two radii, and hence a diameter’s length is twice that of a radius.

A line passing through two points on a circle is a *secant.*

A piece of the circumference is an *arc.*

The area bounded by the circumference and an angle with vertex at the center of the circle is a *sector*.

A tangent line to a circle intersects the circle at only one point. The radius of the circle is perpendicular to the tangent line at the point of tangency:

*AB = AC*

Two tangents to a circle from a common exterior point of the circle are congruent:

An angle inscribed in a semicircle is a right angle:

A central angle has by definition the same measure as its intercepted arc:

An inscribed angle has one-half the measure of its intercepted arc:

The area of a circle is* πr*^{2} , and its circumference (perimeter) is 2*πr*, where *r* is the radius:

*A = πr*^{2}

*C *= 2*πr*

In the GRE, π = 3 is a sufficient approximation for π. You don’t need π = 3.14.

**Example:**

In the figure below, the circle has center O and its radius is 2. What is the length of arc ACB ?

(A) π/3

(B) 2π/3

(C) π

(D) 4π/3

(E) 7π/3

*πr*= 2

*π*(2) = 4

*π*. A central angle has by definition the same degree measure as its intercepted arc. Hence, arc

*ACB*is also 60°. Now, the circumference of the circle has 360°. So arc

*ACB*is $\frac{1}{6}$ (= 60/360) of the circle’s circumference. Hence, arc

*ACB*= $\frac{1}{6}$ (4π) = $\frac{2}{3}$

*π*. The answer is (B).