how to find the length of the arc

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Determining the arc length of a circle is easy with these simple formulas

If you’re learning about arc length in geometry, your teacher probably assigned you a bunch of problems for homework. You have the radius and the central angle of the circle, so how do you find the arc length? Well, you’ve come to the right place! Arc length is the distance between one end of an arc on a circle and the other end of it. In this article, we’ll show you what formulas you need and how to use them to find the arc length of a circle. For more information, read!

[Edit]things you should know

  • When the central angle of a circle is measured in degrees, use the formula arc length = 2π(r)(θ)360){displaystyle {text{arc length}}=2pi (r)({frac {theta} {360}})},
  • If the central angle is in radians, use the formula arc length = θ(r) {displaystyle {text{arc length}}=theta (r)},
  • Plug in the measure of the circle’s radius and the central angle to solve the formula.

[Edit]step

[Edit]Solving when the central angle is in degrees

  1. Set up the formula for the length of the arc. formula is arc length = 2π(r)(θ)360){displaystyle {text{arc length}}=2pi (r)({frac {theta} {360}})}Where? r {displaystyle r} is equal to the radius of the circle and θ{displaystyle theta} Equal to the measure of the central angle of an arc, in degrees.[1]
    Find the Length of the Arc Step 1 Version 3.jpg
  2. Insert the length of the radius of the circle into the formula. This information is usually given to you in a problem. Otherwise, measure the radius of the circle with a ruler or protractor. just substitute the value of radius for the variable r {displaystyle r},
    Find the Length of the Arc Step 2 Version 3.jpg
    • For example, if the radius of the circle is 10 cm, set up the formula as follows: arc length = 2π(10) (θ)360){displaystyle {text{arc length}}=2pi (10)({frac {theta} {360}})},
  3. Insert the value of the central angle of the arc into the formula. Usually, the problem you’re working on provides this information. Be sure to convert the angle to degrees if it’s currently in radians. Then, substitute the measure of the central angle for θ{displaystyle theta} In the formula.
    find the length of the arc step 3 version 3.jpg
    • For example, if the central angle of the arc is 135 degrees, your formula will now look like this: arc length = 2π(10)(135360){displaystyle {text{arc length}}=2pi (10)({frac {135}{360}})},
  4. multiply radius by {displaystyle 2pi}, If You Don’t Have a Calculator, Use an Approximation π=3.14{displaystyle pi =3.14} for your calculations. Rewrite the formula using this new value, which represents the circumference of the circle.[2]
    Find the Length of the Arc Step 4 Version 3.jpg
    • For example, your formula now looks like this:
      arc length = 2π(10)(135360){displaystyle {text{arc length}}=2pi (10)({frac {135}{360}})}
      =2(3.14)(10)(135360){displaystyle =2(3.14)(10)({frac {135}{360}})}
      =(62.8)(135360){displaystyle =(62.8)({frac {135}{360}})}
  5. Divide the central angle of the arc by 360 degrees. Since the sum total of a circle is 360 degrees, dividing the central angle by 360 degrees gives you the part of the circle that represents the area. Using this information, find what part of the circumference the arc length represents.
    Find the Length of the Arc Step 5 Version 2.jpg
    • For example, simplify the formula to get:
      Arc length=(62.8)(135360){displaystyle {text{arc length}}=(62.8)({frac {135}{360}})}
      =(62.8)(.375){displaystyle =(62.8)(.375)}
  6. Multiply the two numbers together. This gives you the length of the arc.
    Find the Length of the Arc Step 6 Version 2.jpg
    • Solve the formula:
      Arc length=(62.8)(.375)=23.55{displaystyle {text{arc length}}=(62.8)(.375)=23.55}
      So, the arc length of a circle with a radius of 10 cm and a central angle of 135 degrees is approximately 23.55 cm.

[Edit]Solving when the central angle is in radians

  1. Set up the formula for the length of the arc. formula is arc length = θ(r) {displaystyle {text{arc length}}=theta (r)}Where? θ{displaystyle theta} is equal to the central angle of the arc in radians, and r {displaystyle r} is equal to the length of the radius of the circle.[3]
    Find the Length of the Arc Step 7 Version 2.jpg
  2. Insert the length of the radius of the circle into the formula. The math problem you’re working on usually provides this information. substitute the length of the radius for the variable r {displaystyle r},
    Find the Length of the Arc Step 8 Version 2.jpg
    • For example, if the radius of the circle is 10 cm, your formula would look like this: arc length = θ(10) {displaystyle {text{arc length}}=theta (10)},
  3. Plug the measure of the arc’s central angle into the formula. When using this formula, the central angle of the arc must be in radians. If the central angle is in degrees, convert it to radians.
    Find the Length of the Arc Step 9 Version 2.jpg
    • For example, if the central angle of the arc is 2.36 radians, your formula will now look like this: arc length=2.36(10){displaystyle {text{arc length}}=2.36(10)},
  4. Multiply the radius by the central angle of the arc. The product gives you the length of the arc.
    find the length of the arc step 10 version 2.jpg
    • For example:
      Arc length=2.36(10)=23.6{displaystyle {text{arc length}}=2.36(10)=23.6}
      So, the arc length of a circle with a radius of 10 cm and a central angle of 23.6 radians is approximately 23.6 cm.

[Edit]Practice Problems and Answers

WH.shared.addScrollLoadItem(‘642bd90345216’)Practice questions and answers to find arc length
WH.shared.addScrollLoadItem(‘642bd90345216’)Practice questions and answers to find arc length

[Edit]Video

[Edit]Advice

  • If you only know the diameter of the circle, divide the diameter by 2 to get the radius. The radius of a circle is half of its diameter.[4] For example, if the diameter of a circle is 14 cm, divide 14 by 2. 14÷2=7{displaystyle 14div 2=7}, Hence, the radius of the circle is 7 cm.

[Edit]Reference

[Edit]quick summary

  1. [v161672_b01], 19 January 2021.
  2. http://mathbitsnotebook.com/Geometry/Circles/CRArcLengthRadian.html
  3. https://mathbitsnotebook.com/Geometry/Circles/CRArcLengthRadian.html
  4. http://www.mathopenref.com/diameter.html

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