How To Identify Arcs, Central Angles, And Lengths Of Arcs - Dummies

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HomeAcademics & The Arts ArticlesMath ArticlesGeometry ArticlesHow to Identify Arcs, Central Angles, and Lengths of ArcsByMark Ryan Updated2021-09-17 16:02:49From the bookGeometry For DummiesShare

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Download E-BookGeometry For Dummies Explore Book Geometry Essentials For Dummies Explore BookBuy NowBuy on AmazonBuy on WileySubscribe on PerlegoDownload E-BookGeometry For DummiesExplore Book Geometry Essentials For DummiesExplore BookBuy NowBuy on AmazonBuy on WileySubscribe on PerlegoA circle's central angles and the arcs that they cut out are part of many circle proofs. They also come up in many area problems. The following figure shows how an angle and an arc are interrelated. A 60-degree central angle cuts out a 60-degree arc.

• Arc: An arc is simply a curved piece of a circle. Any two points on a circle divide the circle into two arcs: a minor arc (the smaller piece) and a major arc (the larger)—unless the points are the endpoints of a diameter, in which case both arcs are semicircles.

Note that to name a minor arc, you use its two endpoints; to name a major arc, you use its two endpoints plus any point along the arc.

• Central angle: A central angle is an angle whose vertex is at the center of a circle. The two sides of a central angle are radii that hit the circle at the opposite ends of an arc—or as mathematicians say, the angle intercepts the arc.

The measure of an arc is the same as the degree measure of the central angle that intercepts it.

Determining the length of an arc

An arc’s length means the same commonsense thing length always means — you know, like the length of a piece of string (with an arc, of course, it’d be a curved piece of string). Make sure you don’t mix up arc length with the measure of an arc which is the degree size of its central angle.

A circle is 360° all the way around; therefore, if you divide an arc’s degree measure by 360°, you find the fraction of the circle’s circumference that the arc makes up. Then, if you multiply the length all the way around the circle (the circle’s circumference) by that fraction, you get the length along the arc. So finally, here’s the formula you’ve been waiting for.

Arc length:

Its degree measure is 45° and the radius of the circle is 12, so here’s the math for its length:

Pretty simple, eh?

About This Article

This article is from the book: 

Geometry For Dummies

About the book author:

Mark Ryan has more than three decades’ experience as a calculus teacher and tutor. He has a gift for mathematics and a gift for explaining it in plain English. He tutors students in all junior high and high school math courses as well as math test prep, and he’s the founder of The Math Center on Chicago’s North Shore. Ryan is the author of Calculus For Dummies, Calculus Essentials For Dummies, Geometry For Dummies, and several other math books.

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