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How to Calculate Arc Length

November 23, 2023 by admin Category: How To

You are viewing the article How to Calculate Arc Length  at Tnhelearning.edu.vn you can quickly access the necessary information in the table of contents of the article below.

The concept of arc length plays a crucial role in various fields, including mathematics, physics, and engineering. Whether you are working on a geometry problem or trying to determine the distance traveled along a curved path, understanding how to calculate arc length is essential. In this guide, we will delve into the intricacies of arc length calculations, exploring different methods and formulas that can be used to find this fundamental measure. By the end, you will have a comprehensive understanding of arc length and the necessary tools to solve problems involving curved objects and paths.

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This article was co-written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical modeling for genome evolution, and data science. Mario holds a bachelor’s degree in mathematics from California State University, Fresno, and a doctorate in applied mathematics from the University of California, Merced. Mario teaches at both the high school and college levels.

This article has been viewed 69,311 times.

An arc is any segment on the circumference of a circle. [1] X Research Source Arc length is the distance from one end of an arc to the other. To find the arc length, you need to have some knowledge of circle geometry. Since the arc is part of the circumference, if you know the angle at the center of the arc it is easy to find out the length of that arc.

Table of Contents

  • Steps
    • Use the measure of the central angle in degrees
    • Use the measure of the central angle in radians
  • Advice

Steps

Use the measure of the central angle in degrees

Image titled Find Arc Length Step 1

Image titled Find Arc Length Step 1

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Create a formula for the arc length. The formula is arc length=2π(r)(θ360){displaystyle {text{arc length}}=2pi (r)({frac {theta }{360}})}{displaystyle {text{arc length}}=2pi (r)({frac {theta }{360}})} , in there r{displaystyle r}r is the radius of the circle and θ{displaystyle theta }theta is the measure of the angle at the center of the arc. [2] X Research Source
Image titled Find Arc Length Step 2

Image titled Find Arc Length Step 2

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Substitute the value of the length of the circle’s radius into the formula. This information must be given a given assignment, or you can measure it. Remember to replace this value in the variable r{displaystyle r}r .

  • For example, if the radius of the circle is 10cm, the formula would look like this: arc length=2π(ten)(θ360){displaystyle {text{arc length}}=2pi (10)({frac {theta }{360}})}{displaystyle {text{arc length}}=2pi (10)({frac {theta }{360}})} .
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Image titled Find Arc Length Step 3

Image titled Find Arc Length Step 3

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Substitute the measure of the angle at the center of the arc into the formula. This information must be given a given assignment, or you can measure it. You must use the angle measure in degrees, not radians, when using this formula. Substitute the measure of the central angle in θ{displaystyle theta }theta in the formula.

  • For example, if the measure of the angle at the center of the arc is 135 degrees, the formula would look like this: arc length=2π(ten)(135360){displaystyle {text{arc length}}=2pi (10)({frac {135}{360}})}{displaystyle {text{arc length}}=2pi (10)({frac {135}{360}})} .
Image titled Find Arc Length Step 4

Image titled Find Arc Length Step 4

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/e/e7/Find-Arc-Length-Step-4-Version-3.jpg/v4-728px-Find-Arc-Length- Step-4-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/e/e7/Find-Arc-Length-Step-4-Version-3.jpg/ v4-728px-Find-Arc-Length-Step-4-Version-3.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:” <div class=”mw-parser-output”></div>”}
Multiply the radius by 2π{displaystyle 2pi }{displaystyle 2pi } . If you are not using a calculator, an approximate value can be used π=3,14{displaystyle pi =3.14}{displaystyle pi =3.14} for calculations. Rewrite the formula with this new value to represent the circumference of the circle. [3] X Research Sources

  • For example:
    2π(ten)(135360){displaystyle 2pi (10)({frac {135}{360}})}{displaystyle 2pi (10)({frac {135}{360}})}
    2(3,14)(ten)(135360){displaystyle 2(3,14)(10)({frac {135}{360}})}{displaystyle 2(3,14)(10)({frac {135}{360}})}
    (62,8)(135360){displaystyle (62.8)({frac {135}{360}})}{displaystyle (62.8)({frac {135}{360}})}
Image titled Find Arc Length Step 5

Image titled Find Arc Length Step 5

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/d/d7/Find-Arc-Length-Step-5-Version-2.jpg/v4-728px-Find-Arc-Length- Step-5-Version-2.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/d/d7/Find-Arc-Length-Step-5-Version-2.jpg/ v4-728px-Find-Arc-Length-Step-5-Version-2.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:” <div class=”mw-parser-output”></div>”}
Divide the angle at the center of the arc by 360. Since a circle has 360 degrees, this calculation will tell you how much of the arc the arc occupies. With this information, you can find out how much of the arc length takes up the circumference.

  • For example:
    (62,8)(135360){displaystyle (62.8)({frac {135}{360}})}{displaystyle (62.8)({frac {135}{360}})}
    (62,8)(0,375){displaystyle (62.8)(0.375)}{displaystyle (62.8)(0.375)}
Image titled Find Arc Length Step 6

Image titled Find Arc Length Step 6

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/c/cf/Find-Arc-Length-Step-6-Version-2.jpg/v4-728px-Find-Arc-Length- Step-6-Version-2.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/c/cf/Find-Arc-Length-Step-6-Version-2.jpg/ v4-728px-Find-Arc-Length-Step-6-Version-2.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:” <div class=”mw-parser-output”></div>”}
Multiply two numbers together. You will find the arc length value.

  • For example:
    (62,8)(0,375){displaystyle (62.8)(0.375)}{displaystyle (62.8)(0.375)}
    23,55{displaystyle 23.55}{displaystyle 23.55}
    So, the arc length of a circle with a radius of 10cm, with a central angle of 135 degrees, is about 23.55 cm.

Use the measure of the central angle in radians

Image titled Find Arc Length Step 7

Image titled Find Arc Length Step 7

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/c/ce/Find-Arc-Length-Step-7-Version-2.jpg/v4-728px-Find-Arc-Length- Step-7-Version-2.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/c/ce/Find-Arc-Length-Step-7-Version-2.jpg/ v4-728px-Find-Arc-Length-Step-7-Version-2.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:” <div class=”mw-parser-output”></div>”}
Create a formula for the arc length. The formula is arc length=θ(r){displaystyle {text{arc length}}=theta (r)}{displaystyle {text{arc length}}=theta (r)} , in there θ{displaystyle theta }theta is the measure of the angle at the center of the arc in radians, and r{displaystyle r}r is the length of the radius of the circle.
Image titled Find Arc Length Step 8

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Image titled Find Arc Length Step 8

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/b/be/Find-Arc-Length-Step-8-Version-2.jpg/v4-728px-Find-Arc-Length- Step-8-Version-2.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/b/be/Find-Arc-Length-Step-8-Version-2.jpg/ v4-728px-Find-Arc-Length-Step-8-Version-2.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:” <div class=”mw-parser-output”></div>”}
Substitute the value of the length of the circle’s radius into the formula. You need to know the length of the radius to be able to use this method. Remember to substitute the length of the radius into the variable r{displaystyle r}r .

  • For example, if the radius of the circle is 10cm, the formula would look like this: arc length=θ(ten){displaystyle {text{arc length}}=theta (10)}{displaystyle {text{arc length}}=theta (10)} .
Image titled Find Arc Length Step 9

Image titled Find Arc Length Step 9

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/7/71/Find-Arc-Length-Step-9-Version-2.jpg/v4-728px-Find-Arc-Length- Step-9-Version-2.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/7/71/Find-Arc-Length-Step-9-Version-2.jpg/ v4-728px-Find-Arc-Length-Step-9-Version-2.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:” <div class=”mw-parser-output”></div>”}
Substitute the measure of the angle at the center of the arc into the formula. You must be given this value in radians. If you know the angle measure in degrees then this method cannot be used.

  • For example, if the measure of the angle at the center of the arc is 2.36 radians, the formula would look like this: arc length=2,36(ten){displaystyle {text{arc length}}=2,36(10)}{displaystyle {text{arc length}}=2,36(10)} .
  • Image titled Find Arc Length Step 10

    Image titled Find Arc Length Step 10

    {“smallUrl”:”https://www.wikihow.com/images_en/thumb/3/33/Find-Arc-Length-Step-10-Version-2.jpg/v4-728px-Find-Arc-Length- Step-10-Version-2.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/3/33/Find-Arc-Length-Step-10-Version-2.jpg/ v4-728px-Find-Arc-Length-Step-10-Version-2.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:” <div class=”mw-parser-output”></div>”}
    Multiply the radius by the radian measure. You will find the arc length value.

    • For example:
      2,36(ten){displaystyle 2,36(10)}{displaystyle 2,36(10)}
      =23,6{displaystyle=23,6}{displaystyle=23,6}
      So, the arc length of a circle with a radius of 10cm, with a central angle of 23.6 radians, is about 23.6 cm.
  • Advice

    • If you know the diameter of the circle it is still possible to find the arc length. In the formula to calculate the arc length there is the radius of the circle. Since the radius is half the diameter, to find the radius you just need to divide the diameter by 2 [4] X Research Source . For example, if the diameter of the circle is 14 cm, you would divide 14 by 2 to get the radius:
      14÷2=7{displaystyle 14div 2=7}{displaystyle 14div 2=7} .
      Therefore, the radius of the circle is 7 cm.
    X

    This article was co-written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical modeling for genome evolution, and data science. Mario holds a bachelor’s degree in mathematics from California State University, Fresno, and a doctorate in applied mathematics from the University of California, Merced. Mario teaches at both the high school and college levels.

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    This article has been viewed 69,311 times.

    An arc is any segment on the circumference of a circle. [1] X Research Source Arc length is the distance from one end of an arc to the other. To find the arc length, you need to have some knowledge of circle geometry. Since the arc is part of the circumference, if you know the angle at the center of the arc it is easy to find out the length of that arc.

    In conclusion, calculating arc length is an important skill in various fields such as mathematics, physics, and engineering. By understanding the concept of arc length and knowing the necessary formulas and techniques, one can accurately determine the distance along the curve of an arc. Whether it is finding the length of a circular track, a bent wire, or the curve of a road, the arc length calculation allows for precise measurements and accurate predictions. Through the use of the central angle, radius, and trigonometric functions, individuals can easily calculate the arc length and apply the results to solve real-life problems. Additionally, technological advancements have simplified the process of arc length calculation, with the availability of computer programs and online arc length calculators. As a fundamental concept in mathematics, mastering arc length calculation enhances one’s problem-solving abilities and opens the door to further exploration and understanding of more complex mathematical concepts. Overall, understanding how to calculate arc length is a valuable skill that plays an integral role in various fields and allows for the analysis and prediction of curved shapes and objects in the physical world.

    Thank you for reading this post How to Calculate Arc Length at Tnhelearning.edu.vn You can comment, see more related articles below and hope to help you with interesting information.

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    1. Formula for calculating arc length
    2. Steps to find arc length
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    4. Finding the central angle for arc length calculation
    5. How to use the arc length formula with radius
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