Circumscribed Angle: Definition & Theorem

Instructor: Matthew Bergstresser
Geometry involves many different shapes. In this lesson, we will deal with a circle and two lines tangent to the circle. This forms a circumscribed angle, which we will learn how to identify and how to calculate its angle.

Ice Cream Cone Geometry

I bet the last thing you thought of the last time you had an ice cream cone was geometry. An ice cream cone is a roughly spherical mass of ice cream on a cone whose sides are tangent to the ice cream. The angle between the two sides of the cone at bottom is called the circumscribed angle. Let's dig deeper into this concept.

The circumscribed angle is between the red tangent lines where they intersect

Circumscribed Angle Theorem

Before we begin discussing the circumscribed angle, we have to draw two tangent lines to a circle. A tangent line is one that touches a curve at only one point. Diagram 1 shows the tangent lines.

Diagram 1. The red lines are tangent lines

Now we can draw two radii from the center of the circle to points A and B on the edge of the circle. Diagram 2 shows the two radii.

Diagram 2

The angle between the radius and the tangent line is 90°. With the addition of these radii to our diagram, we have a quadrilateral, which is a four sided figure. Diagram 3 shows the quadrilateral.

The quadrilateral (shaded) is between points A, P, B, C

The sum of all of the angles in a quadrilateral is 360°. If we know the interior angle between points A and B (we'll call it θ), we can determine the circumscribed angle, which we'll call α. The arc along the circumference of the circle between point A and point B is also equal to θ. Diagram 4 shows the interior angle θ and the arc angle θ.

Diagram 4

Let's make a chart to organize the angles so we can calculate the circumscribed angle.

Angles in Quadrilateral
Angles between tangent lines and radii (two of them) 90°× 2 = 180°
Interior angle θ
Circumscribed angle α

Summing all of these angles, we get

180° + θ + α = 360°

If we knew θ we could calculate α, so let's just estimate what it is. It is definitely larger than 90°, but less than 180°, so well estimate it to be 160°. We will update our chart with this estimated angle for θ.

Angles in Quadrilateral
Angles between tangent lines and radii (two of them) 90°× 2 = 180°
Interior angle 160°
Circumscribed angle α

Now, summing these angles we get

180° + 160° + α = 360°

This means the circumscribed angle, α, equals 20°

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