Inscribed and Central Angles in Circles

Definitions and theorems related to inscribed and central angles in circles are discussed using examples. See also interactive tutorial on central and inscribed angles.



Definitions

1 - A central angle of a circle is an angle whose vertex is located at the center of the circle. Angle BOC in the figure below.

2 - An inscribed angle is an angle whose vertex is on a circle and whose sides each intersect the circle at another point. Angle CAB in the figure below.

definition of central and inscribed angles in a circle.

Theorem

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

angle BAC = (1 / 2) angle BOC


angle BDC = (1 / 2) angle BOC


2 - Two or more inscribed angles intercepting the same arc are equal.

angle BAC = angle BDC


inscribed and central angles intercepting same arc.

Problem 1: In the figure below chord CA has a length of 12 cm. The circle of center O has a radius of 14 cm. Find an approximate value (2 decimal places) to the size of the inscribed angle CBA.

inscribed and central angles problem 1.

Solution to Problem 1:

  • We first calculate the central angle COA. Triangle COA is an isosceles triangle since length of CO = length of AO = radius = 14 cm. We use the cosine law to find cos (angle COA).

    CA 2 = CO 2 + AO 2 - 2 CO AO cos (angle COA)

  • Substitute CA, CO and AO by their numerical values and express cos(angle COA) as follows

    cos(angle COA) = [ 14 2 + 14 2 - 12 2 ] / [2 * 14 * 14 ]

    = 31 / 49

  • Size of angle COA is given by.

    Angle COA = arcos(31 / 49)

  • According to the theorem above, the size of angle CBA is equal to half the size of angle COA.

    Angle CBA = (1 / 2) arcos(31 / 49)

    = 25.38 degrees (approximated to 2 decimal places)

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Updated: 2 April 2013

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