﻿ ﻿Circles - Geometry - Math Review - GMAT Quantitative Review

## 3.0 Math Review

### 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 .

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