How to Determine the Measure of an Angle whose Vertex Is Inside a Circle

By Mark Ryan

An angle that intersects a circle can have its vertex inside, on, or outside the circle. This article covers angles that have their vertex inside a circle—so-called chord-chord angles. The measure of a chord-chord angle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.


For example, check out the above figure, which shows you chord-chord angle SVT. You find the measure of the angle like this:


Look at the following figure:


Here’s a problem to show how the formula plays out:


To use the formula to find angle 1, you need the measures of arcs MJ and KL. You know the ratio of all four arcs is 1 : 3 : 4 : 2, so you can set their measures equal to 1x, 3x, 4x, and 2x. The four arcs make up an entire circle, so they must add up to 360°. Thus,

1x + 3x + 4x + 2x = 360

10x = 360

x = 36


Now use the formula:


That’s a wrap.