﻿ Inscribed Angle of a Circle and the arc it forms. Formula explained with pictures and an interactive demonstration. An Inscribed angle is just... Unfortunately, in the last year, adblock has now begun disabling almost all images from loading on our site, which has lead to mathwarehouse becoming unusable for adlbock users. # Inscribed Angle of a Circle and its intercepted arc

Definition, Formula and Practice

#### What is an Inscribed Angle ?

Is formed by 3 points that all lie on the circle's circumference.

Diagram 1
The Formula

The measure of the inscribed angle is half of measure of the intercepted arc .

$\text{m } \angle b = \frac 1 2 \overparen{AC}$

Explore this relationship in the interactive applet immediately below.

### Interactive Inscribed Angle

$$\angle D = \class{data-angle-0}{35.92} \\ \overparen{\rm BC} = \class{data-angle-1}{35.92}$$
Drag Points To Start Demonstration
Diagram 2

Every single inscribed angle in diagram 2 has the exact same measure, since each inscribed angle intercepts the exact same arc, which is $$\overparen {AZ}$$.

### Practice Problems

##### Problem 1

$$\overparen{XZ}\text{ is the intercepted arc} \\ \angle{Y}\text{ is the inscribed angle} \\$$

If $$m\overparen{XZ} =40^{o}$$, what is the measure of inscribed angle $$Y$$?
$\red{ \text{m } \angle Y} = \frac 1 2 \overparen{XZ} \\ \red{ \text{m } \angle Y } = \frac 1 2 40 ^{\circ} \\ \red{ \text{m } \angle Y } = 20 ^{\circ}$
The error is that $$99^2 + 132^2 \ne 164^2$$. Since the Pythagorean theorem does not hold, $$\angle X$$ is not a right angle and $$\text{ m } \overparen{YWZ} \ne 180^{\circ}.$$