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Inscribed Angle
From Latin: in "inside" + scribere "to write"
Definition: The angle subtended
at a point on the circle by two given points on the circle.
Try this Drag any orange dot. Note that when moving the point P, the inscribed angle is constant
while it is in the
major arc
formed by A,B.
Given two points A and B, lines from them to a third point P form the inscribed angle ∠APB.
As you drag the point P above, notice that the inscribed angle is constant. It only depends on the position of A and B.
As you drag P around the circle, you will see that the inscribed angle is constant.
But when P is in the minor arc (shortest arc between A and B), the angle is still constant, but is the supplement of the usual measure.
That is, it is 180m, where is m is the usual measure.
Formula for inscribed angle
If you know the length of the minor arc and radius, the inscribed angle is given by the formula below.

where:
L is the length of the minor (shortest) arc AB
R is the radius of the circle
π is Pi, approximately 3.142

The formula is correct for points in the major arc.
If the point is in the minor arc, then the will produce the supplement of the correct result, but the the length of the minor arc should still be used in the formula.
Arcs and Chords
The two points A and B can be isolated points, or they could be the end points of an
arc
or
chord.
When they are the end points of an arc, the angle is sometimes called the peripheral angle of the arc.
Central Angle
A similar concept is the central angle. This is the angle subtended at the center of the circle by the two given points.
See Central Angle definition
The central angle is always twice the inscribed angle.
See Central Angle Theorem.
Relationship to Thales' Theorem
Refer to the above figure. If the two points A,B form a diameter of the circle, the inscribed angle will be 90°, which is
Thales' Theorem.
You can verify this yourself by solving the formula above using an arc length of half the circumference of the circle.
You can also move the points A or B above until the inscribed angle is exactly 90°.
You will see that the points A and B are then diametrically opposite each other.
Other circle topics
General
Equations of a circle
Angles in a circle
Arcs
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