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 180-m, where is m is the usual measure.

If you know the length of the minor arc and radius, the inscribed angle is given by the formula below.

where: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.

The central angle is always twice the inscribed angle. See Central Angle 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.

- Circle definition
- Radius of a circle
- Diameter of a circle
- Circumference of a circle
- Parts of a circle (diagram)
- Semicircle definition
- Tangent
- Secant
- Chord
- Intersecting chords theorem
- Intersecting secant lengths theorem
- Intersecting secant angles theorem
- Area of a circle
- Concentric circles
- Annulus
- Area of an annulus
- Sector of a circle
- Area of a circle sector
- Segment of a circle
- Area of a circle segment (given central angle)
- Area of a circle segment (given segment height)

- Basic Equation of a Circle (Center at origin)
- General Equation of a Circle (Center anywhere)
- Parametric Equation of a Circle

- Arc
- Arc length
- Arc angle measure
- Adjacent arcs
- Major/minor arcs
- Intercepted Arc
- Sector of a circle
- Radius of an arc or segment, given height/width
- Sagitta - height of an arc or segment

(C) 2011 Copyright Math Open Reference.

All rights reserved

All rights reserved