Arc Length

    This topic introduces the arc length subtended by a central angle and the area of a wedge cut out by a central angle. Examples are given which emphasize that the central angle must be in radians. Finally, a relationship between arc length and the area of a sector is given.

Proposition (Arc Length) If an angle has its vertex  at the center of a circle (central angle), then the arc length divided by the radius is equal to the number of radians in the angle; that is,

    It is important to remember that can only be used when is in radians.

Example (Arc Length) (a) If a central angle of a diameter circle subtends an arc length of find this angle in degrees.
    Solution. We have where is measured in radians with and Therefore, and so radians. Converting this to degrees we have
    (b) How long will a central angle subtend on a circle 2.5 in. radius?
    Solution. We can use the formula but first we must convert to radians: Therefore, becomes or in.
    (c) If a 30 in diameter bicycle wheel makes 4.0 rev, how far has the bicycle traveled?
    Solution. Again we can use if we first convert to radians: The radius is and so we have

Proposition (Area of a Circular Sector) The area of a circular sector with radius and central angle measured in radians is given by

    It is important to remember that can only be used when is in radians.

Example (Area of a Circular Sector) (a) Find the area of a circular sector having radius and central angle
    Solution. We use the formula where and and thus we have
    (b) A circle 45.9 cm in diameter has a wedge of central angle 134° cut out. What is the area of the remaining portion of the circle?
    Solution. We use the formula where and So, The area of the circle is Therefore the area of the remaining portion of the circle is Alternatively, we can use and so
    (c) If the area of the circular sector is 124 and the central angle is 120°,  find the radius of the circle.
    Solution. We use the formula where and Therefore, and solving for we obtain:
    
    
    
    
    
    
    
    
     
    (d) If the area of a circular sector is and the arc length is , find the radius of the circle and the central angle in degrees.
    Solution. We have and combining these equations together we have Knowing and we can solve for obtaining: and so To find the central angle we can use either or So and thus Finally converting to degrees we have

Definition (Linear Speed) The linear speed is defined as the distance traveled per unit of time that is

Definition (Angular Speed) The angular speed is defined as the amount of rotation per unit of time that is,

Proposition (Relationship between Angular Speed and Linear Speed) If is the linear speed of an object, its angular speed in radians per unit of time, the radius of rotation, then

Example (Relationship between Angular Speed and Linear Speed)  Each tire on an automobile has a radius of 1.25 ft. How many revolutions per minute (rpm) does a tire make when the automobile is traveling at a speed of 88.0 ft/s?
    Solution. We first find the angular speed of a tire by using We have, and so Therefore, the angle through which a tire rotates in one minute is



Since every rad equals one revolution, we have,



Thus the tire is rotating at 672 rpm when the car is traveling at a speed of

• print this page
• pages linking here
• related pages

Cite this as:
Arc Length
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/arc-length.html