Central and Inscribed Angles

Plane Geometry, Central and Inscribed Angles

Central Angle, Inscribed Angle

Any two radii of a circle form a central angle. The angle is proportional to the area enclosed by the angle, which is proportional to the circular arc cut by the angle. In other words, doubling the angle doubles the size of the piece of pie, and doubles the amount of crust at the edge.

Two chords that share an end point form an inscribed angle. The amazing thing is, the inscribed angle is half the central angle.

Let's start with a special case. Let c be the center of the circle. Draw a horizontal diameter from a to b, right through c. Then draw a radius up and to the right, from c to r. Then draw the chord from a to r. The isosceles triangle acr has equal base angles. Let x be the measure of the base angle. Now ∠acr is 180-2x, and the supplementary ∠bcr is 2x. Thus ∠bar is half of ∠bcr. In other words, the inscribed angle is half the central angle.

Add to this picture a radius down and to the right from c to q. Reasoning as above, ∠qab is half of ∠qcb. Add these together and ∠qar is half of ∠qcr. Once again the inscribed angle is half the central angle. This reasoning holds whenever c is inside ∠qar.

Let c lie outside ∠qar. In other words, q and r are both above the diameter ab. As above, ∠baq is half of ∠bcq, and ∠bar is half of ∠bcr. Subtract angles to show that ∠qar is half of ∠qcr. That completes the proof.

c outside inscribed angle