回顾：弧的测量与中心角的测量

(Video) Arcs and Central Angles

by MathHelp.com

This video shows a great step by step solution to a typical central angle problem.

There are some useful formulas to calculate angles in circles, formed by radii, chords, secants, and tangents.

1. Central Angle ========

A central angle is formed by two intersecting radii in a circle. Its vertex must be at the center of the circle.

In this diagram, x is the central angle. In a circle, or congruent circles, congruent central angles have congruent arcs.

Central Angle = Intercepted Arc

Thus,

Angle Measure of Arc AB = Central Angle x (Angle AOB)

1. Inscribed Angle =====

An inscribed angle is an angle with its vertex on the circumference of the circle. It is formed by two intersecting chords.

Inscribed Angle = 1/2 Intercepted Arc

Thus,

1/2*Angle Measure of Arc AC = Angle ABC

1. Tangent Chord Angle =====

An angle formed by an intersecting tangent and chord; the angle has its vertex on the circumference of the circle.

Tangent Chord Angle = 1/2 Intercepted Arc

Angle ABC= 1/2 Measure of Arc AB

1. Angle Formed Inside of a Circle By Two Intersecting Chords ======

When two chords intersect inside a circle, they created two sets of vertical angles (vertical angles are always equal). In the diagram, angle x is on of the vertical angles.

Angle Formed Inside Circle by Intersecting Chords = 1/2 Sum of Intercepted Arcs

Measure of Angle BED = 1/2 (Angle Measure of Arc AC + Angle Measure of Arc BD)

1. Angle Formed By the Intersection of Two Tangents, Two Secants, or a Tangent and a Secant: ========

The formula for ALL THREE of these cases are the same:

Angle Formed Outside Circle = |1/2 Difference of Intercepted Arcs|

a) Two Tangents

Angle ABC is formed by two tangents intersecting outside of the circle at point B. The intercepted arcs are minor arc AC and major arc AC.

b) Two Secants

Angle ACE is formed by two secants intersecting at point C. The intercepted arcs in this diagram are the minor arcs BD and AE.

c) A Secant and a Tangent

Angle ABD is formed by the tangent and secant intersecting at point B. The intercepted arcs in the diagram are the minor arcs AC and AD.

References: "Formulas for Working with Angles in Circles." Geometry Lesson Page. Regentsprep.org, 1988. Web. 6 Sept. 2014.

Text: What is arc measure?

It is important to note that arc measure and arc length are two different concepts. Arc measure is specifically the degree measure of the central angle that defines the arc, while arc length is the length of the arc itself as a segment of the circumference.

A useful consequence of the difference between the two concepts is that you can relate them to one another, or even use them together to find other properties of a circle - like radius, circumference, and area! Read more about properties of circles here.

For example, we know that the two values are both by definition proportional like so: $$\frac{arc\quad measure}{360^\circ}=\frac{arc\quad length}{circumference}$$

Thus, we can find the circumference (and other resulting properties of the circle) through some basic manipulation of the expression above.

Example with Images: Angle Measure and Arc Measure$\\$

Notice that arc length and arc measure are two different things. Arc measure refers to central angel measure.

The arc measure and the are length, however, are related:

Now, Let's look at an example.

Observe the image below. The arc measure is 40 degrees. What is the corresponding arc length?

Using the relationship between arc measure and arc length, we can set up an equation:

Therefore, the arc length is $\frac{4}{9} \pi$.