GMAT Quantitative Review

3.0 Math Review

3.3 Geometry

8. Circles

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).

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 image, where image is approximately image or 3.14. The area of a circle of radius r is equal to image.

image

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

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

image

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

image

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

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.

image

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.

image

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