Circles • Topic 4 of 7

Arc Length

An arc is part of a circle’s circumference, and its length is the matching fraction of the whole circumference. That fraction is the arc’s central angle over 360°, so arc length = (θ/360) × 2πr. A semicircle (180°) is half the circumference; a quarter arc (90°) is one-fourth. For a circle of radius 12 with a 90° arc, the length is (90/360) × 2π(12) = ¼ × 24π = 6π. Keep the answer in terms of π. The key steps are forming the fraction θ/360 and multiplying by the full circumference — confusing arc length (a distance) with sector area (a region) is the usual error.

A circle with two radii forming a 90 degree central angle and the arc between them highlightedArc lengthθArc = (θ/360) × 2πr

✅ Solved examples

1. Radius 12, central angle 90°. Find the arc length in terms of π.
(90/360)(2π·12) = ¼ × 24π = 6π.
2. Radius 9, central angle 180°. Find the arc length.
(180/360)(2π·9) = ½ × 18π = 9π.
3. Radius 6, central angle 60°. Find the arc length.
(60/360)(2π·6) = ⅙ × 12π = 2π.
4. Radius 10, central angle 36°. Find the arc length.
(36/360)(20π) = 0.1 × 20π = 2π.

✏️ Practice — try these, take hints as needed

1. Radius 8, central angle 90°. Find the arc length in terms of π.
(θ/360)(2πr).
¼ × 16π.
—
4π.
2. Radius 15, central angle 180°. Find the arc length.
½ × 30π.
—
—
15π.
3. Radius 12, central angle 60°. Find the arc length.
⅙ × 24π.
—
—
4π.
4. Radius 18, central angle 120°. Find the arc length.
(120/360) × 36π.
⅓ × 36π.
—
12π.
5. Radius 5, central angle 180°. Find the arc length.
½ × 10π.
—
—
5π.

📝 Topic test — 8 questions

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