Arc Length and Sector Area

Geometry · Axiom Academy

LESSON Arc Length and Sector Area Measuring parts of circles using proportions — both the curved arc and the pie-slice are the same fraction of the whole. 1. A Sector Is a Fraction of a Circle Take a slice of a circle cut by a central angle . The slice covers the fraction of the whole turn — and that same fraction governs both the curved arc on the rim and the sector area inside. Arc length = that fraction of the circumference Sector area = that fraction of the total area Multiply the fraction by the whole circumference to get the arc, or by the whole area to get the sector. That's the entire idea. Both quantities are the same fraction of their respective wholes. 3. Example: Finding Arc Length A circle has radius 10 cm. Find the length of an arc with central angle 72° . 4. Example: Finding Sector Area A circle has radius 6 cm. Find the area of a sector with central angle 120° . 5. Bonus: Radian Formulas (Simpler!) When the angle is measured in radians instead of degrees, the bookkeeping disappears and the formulas become much cleaner. Radius times angle (in radians). Half times radius squared times angle. A sector has area cm² and radius 10 cm. Find the central angle. One fraction, , measures both the arc on the rim and the slice inside — and in radians the formulas get even simpler. Scroll up to revisit any step.

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