Arc Measures

It is very important to be familiar with the anatomy of a circle and especially the angles within it. This article covers the properties of arc measures, the formula for an arc measure, and how to find it within a geometric context.

Arc Measures Arc Measures

Create learning materials about Arc Measures with our free learning app!

  • Instand access to millions of learning materials
  • Flashcards, notes, mock-exams and more
  • Everything you need to ace your exams
Create a free account
Contents
Table of contents

    The arc and its measure

    There are two important definitions to be aware of:

    The arc of a circle

    An arc is the edge of a circle sector, i.e. the edge bounded/delimited by two points in the circle.

    Arc length is the size of the arc, i.e. the distance between the two delimiting points on the circle.

    The measure of an arc

    If we think of an arc as being the edge between two points A and B on a circle, the arc measure is the size of the angle between A, the centre of the circle, and B.

    In relation to the arc length, the arc measure is the size of the angle from which the arc length subtends.

    Here are these definitions demonstrated graphically:

    Arc measures finding the measure of an arc StudySmarterFinding the measure of an Arc StudySmarter original

    Radians versus degrees

    Before we introduce the formula for arc measurement, let’s recap degrees and radians.

    To convert radians to degrees: divide by πand multiply by 180.

    To convert degrees to radians: divide by 180 and multiply byπ.

    Here are some of the common angles which you should recognise.

    Degrees030456090120180270360
    Radians0π6π4π3π22π3π3π22π

    Arc measure and arc length formulae

    Finding the arc measure with the radius

    The formula that links both the arc measure (or angle measure) and the arc length is as follows:

    S=r×θ

    Where

    • r is the radius of the circle
    • θ is the arc measure in radians
    • S is the arc length

    We can find the arc measure given the radius and the arc length by rearranging the formula: θ=Sr.

    Find the arc measure shown in the following circle in terms of its radius, r.

    Arc measures Arc measures StudySmarter

    Using the formula S=r×θ:

    13=r×x

    We need the arc measure in terms of r, so we need to rearrange this equation:

    x=13°r

    Finding the arc measure with the circumference

    If we are not given the radius, r, then there is a second method for finding the arc measure. If we know the circumference of a circle as well as the arc length, then the ratio between the arc measure and 360° (or2πc depending on whether you want the arc measure in degrees or radians) is equal to the ratio between the arc length and the circumference.

    θ360°=Sc

    Where

    • c is the circumference of the circle

    • θ is the arc measure in degrees
    • S is the arc length

    Find the arc length, x, of the following circle with a circumference of 10 cm.

    Arc measures finding x StudySmarter

    Using the formula θ2π=Sc:

    5.52π=x10

    Rearranging, we get:

    x=10×5.52×π=8.75 to 3 s.f.

    Arc Measures - Key takeaways

    • An arc is the edge of a circle sector, i.e. the edge bounded/delimited by two points in the circle.
    • Arc length is the size of the arc, i.e. the distance between the two delimiting points on the circle.
    • An arc measure is the size of the angle from which the arc subtends.
    • Finding the arc measure given the radius and arc length:
      • S=r×θ

        Where

        • r is the radius of the circle.

        • θ is the arc measure in radians.
        • S is the arc length.

    • Finding the arc measure given the circumference and arc length:

      • θ360°=Sc

        Where:

        • c is the circumference of the circle.

        • θ is the arc measure in degrees.
        • S is the arc length.

    Frequently Asked Questions about Arc Measures

    What is an arc measure?

    An arc measure is the angle from which an arc of a circle subtends.

    How do you find the measure of an arc?

    How to find the measure of an arc: given the radius and arc length, the arc measure is the arc length divided by the radius. Given the circumference, the ratio between the arc measure and 360 degrees is equal to the ratio between the arc length and the circumference.

    What is the formula for finding the arc measure of an arc?

    The arc measure is the arc length divided by the radius. 

    what is the degree measure of an arc 

    The arc measure is the arc length divided by the radius. 

    what is arc measures geometry with examples 

    In geometry, the arc measure is the arc length divided by the radius. 

    Test your knowledge with multiple choice flashcards

    A line segment that has its endpoints on a circle is called?

    A chord passing through the center of the circle is called?

    Does a chord, apart from a diameter, any other chord splits a circle into a major arc and a minor arc?

    Next

    Discover learning materials with the free StudySmarter app

    Sign up for free
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Math Teachers

    • 4 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App

    Get unlimited access with a free StudySmarter account.

    • Instant access to millions of learning materials.
    • Flashcards, notes, mock-exams, AI tools and more.
    • Everything you need to ace your exams.
    Second Popup Banner