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Inscribed Angle of a Circle

A circle is a very important two-dimensional figure that is probably most commonly seen in geometry. There are a number of concepts and theorem related to a circle. A circle may form many different types of angles within it. The inscribed angle is one of them. This angle is formed by two chords or two secants within a circle. An inscribed angle is formed when two secant lines intersect on a circle. In other words, the inscribed angle in a circle is formed when one of the end points of two chords in a circle meet in a point.

inscribed angle

In the above picture, $\angle$ ABC is the inscribed angle of the circle. Let us go ahead and learn more about inscribed circle, such as - its definition, theorem based on it, example problems on this concept.

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What is Inscribed Angle ?

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The inscribed angle is demonstrated in the following diagram of a circle.

Example on Inscribed Angle

From the picture above,

  • Angle PQR is the inscribed angle.
  • PQ and QR are the chords.
  • Arc PR is the intercepted arc.
  • Formula: Angle PQR = $\frac{Arc PR}{2}$.

Inscribed Angle of a Circle

  • Two or more inscribed angles intercepting a same arc will be equal.

$\angle$ BAC = $\angle$BDC (From above figure)

  • An inscribed angle is the measure of half the central angle intercepting the same arc.

$\angle$ CAB = $\frac{1}{2}$ $\angle$BOC

$\angle$ CDB = $\frac{1}{2}$  $\angle$ BOC

Another Formula for inscribed angle:

If we know the length of the minor arc, radius, the inscribed angle is found by:

Angle=  $\frac{90L}{\pi \times R}$


L is the length of the shortest arc BA

R is defined as radius.

Relationship to Thales' Theorem

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In geometry, according to the Thales' theorem if there are three points of a circle where line joining any two points is a diameter of circle, then angle formed by this diameter with third point on circle is a right angle. This theorem is said to be the special case of inscribed angle theorem.
Have a look at the following diagram :

Thales' Theorem

Given that points A and B are endpoints of a diameter of circle. Thales' theorem states that the inscribed angle would be a right angle in this case. We can prove this in the following way.

Let us suppose that $\angle$CAO = x and $\angle$CBO = y.
So, we would have $\angle$C = x + y
Sum of all three angles of a triangle is 180$^{\circ}$, therefore
x + y + (x + y) = 180$^{\circ}$ 
2(x + y) = 180$^{\circ}$
x + y = 90$^{\circ}$
Hence, it is proved.


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Example 1: From the circle diagram below, the chord CA has a length of 12 cm and center at O. The circle has a radius of 14 cm. Find the measure of the inscribed angle ABC.

Inscribed Angle Examples


1. First we calculate the central angle AOC.The triangle AOC is aisosceles triangle. Distance of OC = Distance of OA = radius = 14 cm. Here we use cosine law to find value of cos (angle AOC).

$AC^2$ = $OC^2 + OA^2 - 2 OC\ OA \cos(\angle COA)$

2. Substitute the value ofthe angles AC, OC and AO in cos (angle AOC) as follows

$cos(\angle COA)$ $\frac{14^2 + 14^2 - 122}{28 \times 14}$


3. The measure of the angle COA is given by.

$\angle COA$ $\cos^{-1}$ $\frac{62}{98}$

According to the theorem described before, the size of angle CBA will be equal to half the size of angle COA.

$\angle CBA$ = $\frac{1}{2}$ $\cos^{-1}$ $\frac{62}{98}$ 

= 25.38 degrees.

Practice Problems

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Problem 1:

Given that the central angle of circle is 50 degrees. Find the angle of inscribed circle.

Answer = 100 degree

Problem 2:

Given that the central angle of circle is 120 degrees. Find the angle of inscribed circle.

Answer = 240 degree.

More topics in Inscribed Angle
Inscribed Angle Theorem
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