## GMAT Quantitative Review

## 3.0 Math Review

### 3.3 Geometry

### 8. Circles

A *circle* is a set of points in a plane that are all located the same distance from a fixed point (the *center* of the circle).

A *chord* of a circle is a line segment that has its endpoints on the circle. A chord that passes through the center of the circle is a *diameter* of the circle. A *radius* of a circle is a segment from the center of the circle to a point on the circle. The words “diameter” and “radius” are also used to refer to the lengths of these segments.

The *circumference* of a circle is the distance around the circle. If *r* is the radius of the circle, then the circumference is equal to , where is approximately or 3.14. The *area* of a circle of radius *r* is equal to .

In the circle above, *O* is the center of the circle and and are chords. is a diameter and is a radius. If , then the circumference of the circle is and the area of the circle is .

The number of degrees of arc in a circle (or the number of degrees in a complete revolution) is 360.

In the circle with center *O* above, the length of arc *RST* is of the circumference of the circle; for example, if , then arc *RST* has length of the circumference of the circle.

A line that has exactly one point in common with a circle is said to be *tangent* to the circle, and that common point is called the *point of tangency*. A radius or diameter with an endpoint at the point of tangency is perpendicular to the tangent line, and, conversely, a line that is perpendicular to a radius or diameter at one of its endpoints is tangent to the circle at that endpoint.

The line above is tangent to the circle and radius is perpendicular to .

If each vertex of a polygon lies on a circle, then the polygon is *inscribed* in the circle and the circle is *circumscribed* about the polygon. If each side of a polygon is tangent to a circle, then the polygon is *circumscribed* about the circle and the circle is *inscribed* in the polygon.

In the figure above, quadrilateral *PQRS* is inscribed in a circle and hexagon *ABCDEF* is circumscribed about a circle.

If a triangle is inscribed in a circle so that one of its sides is a diameter of the circle, then the triangle is a right triangle.

In the circle above, is a diameter and the measure of is .