What is the Relationship Between Arcs and Chords?

In a circle, congruent chords cut off congruent arcs, and a diameter perpendicular to a chord bisects both the chord and its arc. Both facts come from the equal radii that define a circle.

• Arc: An arc is a continuous segment of a circle’s circumference.
• Chord: A chord is a straight line segment whose endpoints lie on the circle. Note: The diameter is the longest chord of a circle.

• Chords that are equidistant from the center of a circle are equal in length.
• Equal chords of a circle subtend equal angles at the center.
• The perpendicular bisector of a chord passes through the circle’s center.

• Equal chords intercept equal arcs.
• The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

Worked examples: arcs and chords

Practice questions on arcs and chords

1. In a circle with a radius of (7 text{ cm}), what is the approximate length of a chord that intercepts an arc of (90^circ)?
2. If an arc subtends an angle of (30^circ) at the boundary of a circle, what angle does it subtend at the center?

1. ( c approx 2 times 7 times sin(45^circ) approx 9.9 text{ cm})
2. ( 2 times 30^circ = 60^circ )

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Arc vs chord at a glance

Every chord cuts off an arc, and in the same circle equal chords always cut off equal arcs. The chord is always the shorter path, because a straight line is the shortest way between two points.

Worked example: arc length and chord length for the same angle

A circle has radius 7 cm. Two radii make a central angle of 90°. Find the arc length and the chord length between the two endpoints.

1. Arc length. The arc is 90/360 = ¼ of the circle. The full circumference is 2π × 7 = 14π. So the arc is ¼ × 14π = 3.5π ≈ 11.0 cm.
2. Chord length. Draw a segment from the center that is perpendicular to the chord. It cuts the chord and the 90° angle in half. Each half of the chord is 7·sin(45°) ≈ 4.95 cm, so the whole chord is 2 × 7 × sin(45°) ≈ 9.9 cm.
3. Compare. The arc is about 11.0 − 9.9 = 1.1 cm longer than the chord.

As the angle gets smaller, the arc and chord get closer in length. As the angle grows toward 180°, the chord becomes the diameter and the arc becomes a semicircle.

Frequently Asked Questions

What is the relationship between chords and arcs?

Every chord cuts off an arc with the same two endpoints, and the two are linked by the central angle θ. Arc length is (θ/360) × 2πr and chord length is 2r·sin(θ/2). In a circle of radius 6 with θ = 90°, the arc is 3π ≈ 9.42 while the chord is 12 × sin 45° ≈ 8.49. The chord is always shorter because it takes the straight path.

What is the relationship between congruent chords and their arcs?

In the same circle, or in congruent circles, congruent chords cut off congruent arcs, and congruent arcs have congruent chords. This works because equal chords form equal central angles. So if two 8 cm chords sit in one circle and the minor arc of the first measures 70°, the minor arc of the second also measures 70°.

What is the relationship between an arc and an angle?

The measure of an arc equals the measure of its central angle, the angle formed at the center by the two radii to the arc’s endpoints. An inscribed angle that intercepts the same arc measures half of it. So for a 100° arc, the central angle is 100° and any inscribed angle standing on that arc is 50°.

What is the relationship between arcs created between parallel chords?

Two parallel chords in a circle cut off congruent arcs between them. If chords AB and CD are parallel, with A next to C and B next to D, then arc AC and arc BD have equal measure. For example, if arc AC measures 40°, arc BD also measures 40°. The same rule holds when a tangent line is parallel to a chord.