DWG NO. 12.1 — Lesson 1 of 5

Circle Vocabulary and Central Angles

Unit 12: Circles · ~15–30 min

Objective

Identify the parts of a circle and find the measure of a central angle or its intercepted arc.

Defining a circle

A circle is the set of all points in a plane that are the same distance — the radius — from a fixed point called the center. Everything in this unit follows from that single idea: one center, one distance, and every point that satisfies it.

Parts of a circle

Term	Definition
Center	The fixed point every point on the circle is equidistant from. Circles are named by their center (circle O).
Radius	A segment from the center to any point on the circle. All radii of the same circle are congruent.
Diameter	A chord that passes through the center; its length is twice the radius (d = 2r).
Chord	A segment whose endpoints both lie on the circle. A diameter is the longest possible chord.
Central angle	An angle whose vertex is the center of the circle and whose sides are two radii.
Arc	A piece of the circle between two points on it, measured in degrees.

Central angles and their arcs

Every central angle "cuts off" or intercepts an arc, and the two share the same measure — a central angle of 70° always intercepts an arc of 70°. Arcs come in three flavors:

Minor arc — smaller than a semicircle (< 180°), named by its two endpoints, e.g. arc AB.
Major arc — larger than a semicircle (> 180°), named with three points so it's unambiguous, e.g. arc ACB. Its measure is 360° minus the minor arc.
Semicircle — exactly half the circle (180°), cut off by a diameter.

Arc Addition Postulate: the measure of an arc formed by two adjacent arcs is the sum of the two arc measures — the same idea as the Angle Addition Postulate, just wrapped around a circle.

Central angle AOB measures 70°, so its intercepted minor arc AB also measures 70°.

Worked Example 1 · Minor and major arc

ProblemIn circle O, central angle AOB measures 84°. Find the measure of minor arc AB and major arc AB.

1.A central angle and its intercepted (minor) arc always have equal measure, so minor arc AB = 84°.

2.The major arc is the rest of the circle: 360° − 84° = 276°.

Minor arc AB = 84°; major arc AB = 276°

Worked Example 2 · Arc addition

ProblemPoints A, B, and C lie on circle O in that order. Arc AB = 3x°, arc BC = 5x°, and arc AC (the major arc, going the other way through the top) is a semicircle’s worth combined with AB and BC to complete the circle: arc AB + arc BC + arc CA = 360°, where arc CA = 130°.

1.Substitute: 3x + 5x + 130 = 360.

2.Combine and solve: 8x + 130 = 360, so 8x = 230, x = 28.75.

3.Then arc AB = 3(28.75) = 86.25° and arc BC = 5(28.75) = 143.75°.

x = 28.75; arc AB = 86.25°, arc BC = 143.75°

Guided practice

1.In circle P, central angle MPN measures 55°. What is the measure of arc MN (minor arc)?
Show answer
A central angle and its intercepted arc are equal, so arc MN = 55°.
2.Using the same circle P from problem 1, what is the measure of the major arc MN?
Show answer
360° − 55° = 305°.
3.Diameter segment AC passes through center O. What is the measure of each semicircle it creates, and why?
Show answer
180° each. A diameter splits the circle into two equal halves, and a full circle is 360°, so each semicircle is 360° ÷ 2 = 180°.
4.Points X, Y, Z lie on circle O in that order, dividing it into three arcs. Arc XY = 110° and arc YZ = 95°. Find arc ZX.
Show answer
By the Arc Addition Postulate, all three arcs sum to 360°: arc ZX = 360 − 110 − 95 = 155°.
5.A student says, "A chord and a diameter are basically the same thing." Explain what's wrong with that statement.
Show answer
A chord is any segment with both endpoints on the circle — a diameter is a special chord that also passes through the center. Every diameter is a chord, but most chords are not diameters (they don't pass through the center and aren't the longest possible chord).

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