Geometry Tool Belt: Inscribed Angles

Geometry Tool Belt: Inscribed Angles

Author: c o

To learn the formula relating arc lengths to inscribed angles, and to see a graphical examples.

To see and understand the proof of this formula.

Necessary background information is listed and briefly reviewed. The formula relating arc length and inscribed angles is given, and then the proof is presented in a three part video series.

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Before beginning this lesson, you should have learned about the following:

  • The interior angles of a triangle add up to 180 degrees
  • The definition of a circle, an arc, a central angle, an arc length, and an inscribed angle.


  • The measure of an arc length is the measure of its central angle
  • The definition of supplementary angles: angles A and B are supplementary when A+B = 180 degrees. 

Inscribed Angles and Arc Lengths

Given a circle with an inscribed angle like  with inscribed angle a and arc b, what is the relation ship between a and b?

The formula relating these two values is

That is, the measure of an inscribed angle is half the measure of the arc its endpoints define.  

So or example, if we have an inscribed angle with of 40 degrees, then the arc length is 80 degrees.

Likewise if we're given an arc length of 50 degrees, then any inscribed angle with line segments intersecting the end points of the arc, where ever its vertex happens to be on the circle, will have an angle measure of 25 degrees.

Now that we know the formula, we should know how it came about.  Working through the proof of this relationship will complete your understanding of the concept, and exercise your knowlege in the geometry you've learned so far.

Proving the Formula Case 1 - When a line segment is the diameter

This video starts off the three part proof by treating the special case of an inscribed angle with one line segment along the circle's diameter.

Source: youtube, Colin O'Keefe

Case 2: When the center is between the segments

This is the second part of our proof where the formula is derived for the case when the center of the circle falls between the two line segments that define the inscribed angle.

Source: youtube, Colin O'Keefe

Case 3 - when the circle center is not between the line segments

This is the third and final portion of the proof. The video derives the formula for the case when the circle's center does not fall between the line segments defining the inscribed angle.

Source: Colin O'Keefe