arc measure — Definition, Formula & Examples
Arc measure is the number of degrees an arc spans along a circle. It equals the measure of the central angle that intercepts (cuts off) that arc.
The arc measure of an arc on a circle is the measure, in degrees, of the central angle that subtends the arc, where is the center of the circle. A minor arc has a measure between and , a semicircle measures exactly , and a major arc measures between and .
How It Works
To find an arc measure, identify the central angle that intercepts the arc — the arc measure equals that angle in degrees. If you know only the major arc, subtract it from to get the minor arc measure, and vice versa. Arc measure is not a length; it tells you what fraction of the full circle the arc occupies. Two arcs can have the same arc measure but very different arc lengths if their circles have different radii.
Worked Example
Problem: In a circle with center O, central angle AOB measures 110°. Find the arc measure of minor arc AB and major arc AB.
Minor arc measure: The minor arc measure equals the central angle.
Major arc measure: The major arc is the rest of the circle, so subtract from 360°.
Answer: Minor arc AB measures 110° and major arc AB measures 250°.
Why It Matters
Arc measure connects directly to inscribed angle theorems and sector area calculations in geometry courses. In trigonometry, converting arc measures to radians is a foundational skill for working with unit-circle definitions of sine and cosine.
Common Mistakes
Mistake: Confusing arc measure (degrees) with arc length (a distance).
Correction: Arc measure is an angle in degrees. Arc length is a physical distance calculated by . Two arcs on different-sized circles can share the same arc measure yet have very different lengths.