central angle

Angle between two chords

Problem
Chords AB and AC are drawn on a circle of radius 10 inches. Find the angle between the chords if the arc BAC is 28 inches long.
 

009-chords-and-arc.gif

 

Quadrilateral with one side as diameter of circumscribing circle

Problem PG-010
The quadrilateral ABCD shown in Fig. PG-010 is inscribed in a circle with side AD coinciding with the diameter of the circle. if sides AB, BC, and CD are 8 cm, 10 cm, and 12 cm long, respectively, find the radius of the circumscribing circle.
 

Cyclic quadrilateral inscribed in a circle of unknown radius

 

Area of Regular Six-Pointed Star

Problem
Find the area of the regular six-pointed star inscribed in a circle of radius 20 cm.
 

Area of Regular Five-Pointed Star

Problem
Find the area of the regular five-pointed star inscribed in a circle of radius 20 cm.
 

Relationship Between Central Angle and Inscribed Angle

Central angle = Angle subtended by an arc of the circle from the center of the circle.
Inscribed angle = Angle subtended by an arc of the circle from any point on the circumference of the circle. Also called circumferential angle and peripheral angle.
 

Figure below shows a central angle and inscribed angle intercepting the same arc AB. The relationship between the two is given by
 

$\alpha = 2\theta \, \text{ or } \, \theta = \frac{1}{2}\alpha$

 

if and only if both angles intercepted the same arc. In the figure below, θ and α intercepted the same arc AB.
 

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