Arc Length and Sector Area

Circle Sectors and Arcs

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Arc length and sector area connect angle measure to parts of a circle. They let you find how far along the edge of a circle you travel and how much of the circle's interior is covered by a slice. These ideas appear in wheel motion, clock hands, gears, and circular design. Learning them helps students move between geometry formulas, proportional reasoning, and real applications.

Both arc length and sector area depend on what fraction of the full circle is selected by the central angle. If the angle is measured in degrees, you compare it to 360 degrees. If the angle is measured in radians, the formulas become especially compact and useful. Understanding when to use degrees, radians, radius, and diameter correctly is the key to solving these problems accurately.

Key Facts

Arc length in degrees: $s = \frac{\theta}{360} \times 2\pi r$
Sector area in degrees: $A = \frac{\theta}{360} \times \pi r^2$
Arc length in radians: $s = r\theta$
Sector area in radians: $A = \frac{1}{2} r^2 \theta$
Circumference of a circle: $C = 2\pi r$
Area of a circle: $A = \pi r^2$

Vocabulary

Arc: An arc is a portion of a circle's circumference between two points.
Sector: A sector is the region inside a circle bounded by two radii and the included arc.
Central angle: A central angle is an angle whose vertex is at the center of the circle.
Radius: A radius is a segment from the center of a circle to any point on the circle.
Radian: A radian is an angle measure based on the ratio of arc length to radius.

Common Mistakes to Avoid

Using the diameter instead of the radius, which gives answers that are too large or too small because all arc length and sector area formulas here are written in terms of r.
Mixing degree formulas with radian angles, which is wrong because $s = r\theta$ and $A = \frac{1}{2}r^2\theta$ only work when $\theta$ is in radians.
Forgetting that arc length is linear and sector area is square units, which leads to incorrect units such as $\text{cm}^2$ for an arc or cm for an area.
Using the fraction $\frac{\theta}{180}$ instead of $\frac{\theta}{360}$ for a sector, which is wrong because a full circle is 360 degrees, not 180 degrees.

Practice Questions

1 A circle has radius 8 cm and central angle 135 degrees. Find the arc length of the sector in terms of $\pi$ .
2 A sector has radius 10 m and central angle 1.2 radians. Find its area.
3 Two sectors are cut from different circles, and both have the same central angle in radians. One circle has twice the radius of the other. Explain how the arc lengths compare and how the sector areas compare.