Circles in Geometry

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A series of free, online High School Geometry Lessons with examples and solutions.
Videos, worksheets, and activities to help Geometry students.

In this lesson, we will learn

  • cyclic quadrilaterals and parallel lines in circles
  • how to calculate the circumference of a circle
  • how to calculate the arc length of a circle
  • about the length of a secant

Cyclic Quadrilaterals and Parallel Lines in Circles

A cyclic quadrilateral has vertices on the same circle and is inscribed in the circle. The opposite angles have the same endpoints (the other vertices) and together their intercepted arcs include the entire circle. Since the measure of an inscribed angle is half the intercepted arc, the sum of the opposite angles must be 180 degrees.

How to prove that opposite angles in a cyclic quadrilateral are congruent?
How to prove that parallel lines create congruent arcs in a circle?

In this lesson we looked at properties of cyclic quadrilaterals


Circumference can be thought of as the “perimeter” of a circle or the distance around a circle. Since pi is the ratio of circumference to diameter, circumference can be calculated by multiplying the circle’s diameter by pi. Another formula substitutes d = 2r, where one diameter equals two radii, and C = 2(r)(pi). Related topics include area of a circle, arc length, and parts of a circle.

How to calculate the circumference of a circle?

This video gives several examples of the process used to find the circumference of a circle.

Arc Length

Commonly confused with arc measure, arc length is the distance between the endpoints along the circle. Arc measure is a degree measurement, equal to the central angle that forms the intercepted arc. Arc length is a fraction of the circumference of the circle and calculated that way: find the circumference of the circle and multiply by the measure of the arc divided by 360.

How to define arc length and how it is different from arc measure; how to calculate the length of an arc?

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.


A secant is a line, ray, or line segment that intersects a circle in two places. Three points are covered:
(1) secants that intersect in a circle which divide each other proportionally.
(2) the angle formed by secants which intersects in a circle and is half the sum of the intercepted arcs.
(3) two secants drawn from the same point outside a circle that form an angle whose measure is half the difference of the intercepted arcs.

How to define a secant; how to find the length of various secants in circles?

Segments of secants theorem.
Two secant segments which share an endpoint outside of the circle. The product of one secant segment and its external segment is equal to the product of the other secant segment and its external segment.

The segments of a secant segment and a tangent segment which share an endpoint outside of the circle. The product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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