Circles - GRE Math
In these lessons, we will learn how to find the circumference of a circle, arc length, area of a circle, area of sector, tangent of a circle, inscribed and crcumscribed polygons.
Circles: Radius, Diameter and Circumference
Given a point O in a plane and a positive number r, the set of points in the plane that are a distance of r units from O is called a circle
. The point O is called the center of the circle and the distance r is called the radius
of the circle. The diameter
of the circle is twice the radius. Two circles with equal radii are
called congruent circles
. Two or more circles with the same center are called concentric circles
Any line segment joining two points on the circle is called a chord
. The terms “radius” and “diameter” can also refer to line segments: A radius is any line segment joining a point on the circle and the center of the circle, and a diameter is any chord that passes through the center of the circle.
The distance around a circle is called the circumference
of the circle, which is analogous to the perimeter of a polygon. The ratio of the circumference C to the diameter d is the same for all circles and is denoted by the Greek letter π (pi).
This video will help you understand the relationship between the radius diameter and circumference of a circle.
C = πd
Central Angles, Arcs and Chords
A central angle
of a circle is an angle with its vertex at the center of the circle. Given any two points on a circle, an arc
is the part of the circle containing the two points and all the points between them. The measure of an arc is the measure of its central angle, which is the angle formed by two radii that connect the center of the circle to the two endpoints of the arc.
In this video, you will learn about Central Angles and their relationship to Arcs. You will also learn about Chords and their relationships to Arcs and Central Angles.
To find the length of an arc of a circle, note that the ratio of the length of an arc to the circumference is equal to the ratio of the degree measure of the arc to 360°.
This video lesson discusses how to find the length of an arc. First, the arc length theorem is reviewed and explained. An example of find the length of a major arc is modeled. The given information is the measure of the related minor arc and the radius of the circle.
Area of Circle
The area of a circle with radius r is equal to πr2
. For example, the area of a circle with radius 6 is π62
This video shows how to get the area of a circle and how it relates to radius and diameter.
Area of Sector
A sector of a circle is a region bounded by an arc of the circle and two radii. Tthe ratio of the area of a sector of a circle to the area of the entire circle is equal to
the ratio of the degree measure of its arc to 360°.
This video shows how to find the area of a sector.
Tangent to a Circle
A tangent to a circle is a line that intersects the circle at exactly one point, called the point of tangency. If a line is tangent to a circle, then a radius drawn to the point of tangency is perpendicular to the tangent line. The converse is also true; that is, if a line is perpendicular to a radius at its endpoint on the circle, then the line is a tangent to the circle at that endpoint.
This video provides example problems of determining unknown values using the properties of a tangent line to a circle.
Inscribed and Circumscribed Polygon
A polygon is inscribed in a circle if all its vertices lie on the circle, or equivalently, the circle is circumscribed about the polygon. If one side of an inscribed triangle is a diameter of the circle then the triangle is a right triangle. Conversely, if an inscribed triangle is a right triangle, then one of its sides is a diameter of the circle.
A polygon is circumscribed about a circle if each side of the polygon is tangent to the circle, or equivalently, the circle is inscribed in the polygon.
This video shows how to solve problems involving quadrilaterals inscribed in circles.
This video shows that if an inscribed triangle is a right triangle, then the hypotenuse is the diameter. If an inscribed angle has a diameter as one of its sides, then its a right triangle.
Tjis video gives a lesson on polygons inscribed in and circumscribed about a circle.
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