Geometry · Lesson 7

Circles: Arcs, Chords & Angles

A circle is defined by its center and radius. Inside that simple shape live powerful theorems connecting angles, arcs, and chords that appear everywhere from engineering to architecture.

Circle Vocabulary

Center (O): The point equidistant from all points on the circle.
Radius (r): Distance from center to any point on the circle.
Diameter (d): Longest chord; d = 2r. Passes through the center.
Chord: A line segment with both endpoints on the circle.
Arc: A portion of the circle's circumference. Minor arc < 180°; major arc > 180°.
Sector: A "pie slice" region bounded by two radii and an arc.
Tangent: A line that touches the circle at exactly one point (perpendicular to radius at that point).
Secant: A line that intersects the circle at two points.

Central Angle Theorem

A central angle has its vertex at the center of the circle. Its measure equals the measure of the intercepted arc:

∠AOB = arc AB

Inscribed Angle Theorem

An inscribed angle has its vertex on the circle and its sides are chords. It equals half the intercepted arc:

Inscribed angle = ½ × intercepted arc

Corollary: All inscribed angles that intercept the same arc are equal. An inscribed angle in a semicircle (intercepting a diameter) is always 90°.

Chord Properties

Two chords that are equal in length are equidistant from the center.
The perpendicular bisector of any chord passes through the center.
When two chords intersect inside a circle: segment × segment = segment × segment (intersecting chords theorem).

Arc Length & Sector Area

For a circle with radius r and central angle θ (in degrees):

Arc Length: L = (θ/360) × 2πr
Sector Area: A = (θ/360) × πr²

Key Insight: The inscribed angle theorem is the foundation for many circle proofs. When an inscribed angle equals 90°, the chord it intercepts must be a diameter — a powerful test for whether a chord is actually the longest possible chord.

Quick Check

1. A central angle of 120° intercepts an arc. What is the arc measure?

60°

120°

240°

180°

2. An inscribed angle intercepts an arc of 80°. What is the inscribed angle?

80°

40°

160°

20°

3. An inscribed angle intercepts a diameter. The inscribed angle measures:

45°

180°

90°

60°

4. The perpendicular bisector of any chord in a circle always passes through the:

Midpoint of another chord

Center of the circle

Tangent point

Circumscribed vertex

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