phd

geometry assignment find the length of each arc

Arc Length Calculator

Table of contents

This arc length calculator is a tool that can calculate the length of an arc and the area of a circle sector. This article explains the arc length formula in detail and provides you with step-by-step instructions on how to find the arc length. You will also learn the equation for sector area.

In case you're new to circles, calculating the length and area of sectors could be a little advanced, and you need to start with simpler tools, such as circle length and circumference and area of a circle calculators.

Arc length formula

The length of an arc depends on the radius of a circle and the central angle θ . We know that for the angle equal to 360 degrees (2π), the arc length is equal to circumference . Hence, as the proportion between angle and arc length is constant, we can say that:

L / θ = C / 2π

As circumference C = 2πr ,

L / θ = 2πr / 2π

L / θ = r We find out the arc length formula when multiplying this equation by θ:

Hence, the arc length is equal to radius multiplied by the central angle (in radians).

Area of a sector of a circle

We can find the area of a sector of a circle in a similar manner. We know that the area of the whole circle is equal to πr². From the proportions,

A / θ = πr² / 2π

A / θ = r² / 2

The formula for the area of a sector is:

A = r² × θ / 2

How to find the length of an arc and sector area: an example

Decide on the radius of your circle. For example, it can be equal to 15 cm. (You can also input the diameter into the arc length calculator instead.)

What will be the angle between the ends of the arc? Let's say it is equal to 45 degrees, or π/4.

Calculate the arc length according to the formula above:

L = r × θ = 15 × π/4 = 11.78 cm

Calculate the area of a sector:

A = r² × θ / 2 = 15² × π/4 / 2 = 88.36 cm²

You can also use the arc length calculator to find the central angle or the circle's radius. Simply input any two values into the appropriate boxes and watch it conducting all calculations for you.

Make sure to check out the equation of a circle calculator , too!

How do you find arc length without the radius?

To calculate arc length without radius, you need the central angle and the sector area :

  • Multiply the area by 2 and divide the result by the central angle in radians.
  • Find the square root of this division.
  • Multiply this root by the central angle again to get the arc length.
  • The units will be the square root of the sector area units.

Or the central angle and the chord length :

  • Divide the central angle in radians by 2 and perform the sine function on it.
  • Divide the chord length by double the result of step 1. This calculation gives you the radius.
  • Multiply the radius by the central angle to get the arc length.

How do you find arc length using radians?

  • Multiply the central angle in radians by the circle’s radius.
  • That’s it! The result is simply this multiplication.

How do you calculate arc length without the angle?

To calculate arc length without the angle, you need the radius and the sector area :

  • Multiply the area by 2.
  • Then divide the result by the radius squared (make sure that the units are the same) to get the central angle in radians.

Or you can use the radius and chord length :

  • Divide the chord length by double the radius.
  • Find the inverse sine of the result (in radians).
  • Double the result of the inverse sine to get the central angle in radians.
  • Once you have the central angle in radians, multiply it by the radius to get the arc length.

Does arc length have to be in radians?

Arc length is a measurement of distance, so it cannot be in radians . The central angle, however, does not have to be in radians. It can be in any unit for angles you like, from degrees to arcsecs. Using radians , however, is much easier for calculations regarding arc length, as finding it is as easy as multiplying the angle by the radius.

© Omni Calculator

Central angle (θ)

Sector area (A)

Chord length (c)

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ol{padding-top:0px;}.css-4okk7a ul:not(:first-child),.css-4okk7a ol:not(:first-child){padding-top:4px;} Arc length

Arc length (L)

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2 Different Ways to Find a Circle’s Arc Length

Last Updated: April 25, 2023 Fact Checked

Solving When the Central Angle is in Degrees

Solving when the central angle is in radians, practice problems and answers, expert q&a.

This article was co-authored by Mario Banuelos, PhD and by wikiHow staff writer, Devin McSween . 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 models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,088,440 times.

If you’re learning about arc lengths in geometry, your teacher probably just assigned you a bunch of problems for homework. You’ve got the circle’s radius and the central angle, so how do you find the length of the arc? Well, you’ve come to the right place! Arc length is the distance between one endpoint of an arc on a circle to the other. In this article, we’ll tell you what formulas you need and how to use them to find a circle's arc length. Read on to learn more!

Things You Should Know

{\text{arc length}}=2\pi (r)({\frac {\theta }{360}})

  • Plug in the circle’s radius and the central angle’s measurement to solve either formula.

Step 1 Set up the formula for arc length.

14\div 2=7

You Might Also Like

Calculate the Circumference of a Circle

  • ↑ Mario Banuelos, PhD. Assistant Professor of Mathematics. Expert Interview. 19 January 2021.
  • ↑ http://mathbitsnotebook.com/Geometry/Circles/CRArcLengthRadian.html
  • ↑ ​​ https://mathbitsnotebook.com/Geometry/Circles/CRArcLengthRadian.html
  • ↑ http://www.mathopenref.com/diameter.html

About This Article

Mario Banuelos, PhD

To find arc length, start by dividing the arc's central angle in degrees by 360. Then, multiply that number by the radius of the circle. Finally, multiply that number by 2 × pi to find the arc length. If you want to learn how to calculate the arc length in radians, keep reading the article! Did this summary help you? Yes No

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Arc Length Calculator

  • degrees(deg)
  • radians(rad)
  • gradians(gon)
  • minutes of arc(arcmin)
  • seconds of arc(arcsec)
  • milliradians(mrad)
  • millimeters(mm)
  • centimeters(cm)
  • kilometers(km)
  • square millimeters(mm²)
  • square centimeters(cm²)
  • square decimeters(dm²)
  • square meters(m²)
  • square kilometers(km²)
  • square inches(in²)
  • square feet(ft²)
  • square yards(yd²)
  • square miles(mi²)

To use the arc length calculator, simply enter the central angle and the radius into the top two boxes. If we are only given the diameter and not the radius we can enter that instead, though the radius is always half the diameter so it’s not too difficult to calculate. 

The calculator will then determine the length of the arc . It will also calculate the area of the sector with that same central angle.

How to Calculate the Area of a Sector and the Length of an Arc

Our calculators are very handy, but we can find the arc length and the sector area manually. It’s good practice to make sure you know how to calculate these measurements on your own.

geometry assignment find the length of each arc

How to Find the Arc Length

An arc length is just a fraction of the circumference of the entire circle. So we need to find the fraction of the circle made by the central angle we know, then find the circumference of the total circle made by the radius we know. Then we just multiply them together. Let’s try an example where our central angle is 72° and our radius is 3 meters. 

First, let’s find the fraction of the circle’s circumference our arc length is. The whole circle is 360°. Let’s say our part is 72°. We make a fraction by placing the part over the whole and we get \(\frac{72}{360}\) , which reduces to \(\frac{1}{5}\) . So, our arc length will be one fifth of the total circumference. Now we just need to find that circumference.

The circumference can be found by the formula C = πd when we know the diameter and C = 2πr when we know the radius, as we do here. Plugging our radius of 3 into the formula, we get C = 6π meters or approximately 18.8495559 m.

Now we multiply that by \(\frac{1}{5}\)  (or its decimal equivalent 0.2) to find our arc length, which is 3.769911 meters. Note that our units will always be a length.

How to Find the Sector Area

Just as every arc length is a fraction of the circumference of the whole circle, the sector area is simply a fraction of the area of the circle. So to find the sector area, we need to find the fraction of the circle made by the central angle we know, then find the area of the total circle made by the radius we know. Then we just multiply them together. Let’s try an example where our central angle is 72° and our radius is 3 meters.

First, let’s find the fraction of the circle’s area our sector takes up. The whole circle is 360°. Our part is 72°. We make a fraction by placing the part over the whole and we get \(\frac{72}{360}\) , which reduces to \(\frac{1}{5}\) . So, our sector area will be one fifth of the total area of the circle. Now we just need to find that area.

The area can be found by the formula A = πr 2 . Plugging our radius of 3 into the formula we get A = 9π meters squared or approximately 28.27433388 m 2 . 

Now we multiply that by \(\frac{1}{5}\)  (or its decimal equivalent 0.2) to find our sector area, which is 5.654867 meters squared.  Note that our answer will always be an area so the units will always be squared.

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Arc of a Circle Calculator

Whether you need to calculate the length of an arc, chord length, or other properties of a circle, our calculator is here to make your life easier.

How to use the Calculator:

  • Enter the diameter to calculate the radius or input the radius to find the diameter.
  • Enter either the angle of the arc or the length of the arc.

By performing these operations, you can quickly find the Arc of a Circle.

But that's not all! Our Arc of a Circle Calculator can also help you:

  • Find the radius of a circle, knowing only the diameter.
  • Estimate the diameter of a circle when its radius is known.
  • Find the length of an arc, using the chord length and arc angle.
  • Compute the arc angle by inserting the values of the arc length and radius.

Arc of a Circle Calculator

Radius (r):

Diameter (d):

Central Angle (θ): °

Arc Length (L):

Chord Length (c):

This calculator uses the following formulas:

Radius = Diameter / 2

Arc length = 2 × π × Radius × (Central Angle [degrees] / 360)

Chord length = 2 × Radius × sin(Central Angle [degrees] / 2)

Where π is the constant (3.141592654)

  • Currently 3.81/5

Rating: 3.8 /5 (209 votes)

geometry assignment find the length of each arc

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High school geometry

Course: high school geometry   >   unit 8.

  • Intro to arc measure
  • Finding arc measures
  • Arc measure

Finding arc measures with equations

  • Arc measure with equations

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Video transcript

IMAGES

  1. How to Find Arc Length: 10 Steps (with Pictures)

    geometry assignment find the length of each arc

  2. How to Calculate Length of an Arc

    geometry assignment find the length of each arc

  3. Arc Length

    geometry assignment find the length of each arc

  4. Arc Length

    geometry assignment find the length of each arc

  5. Unit 11 Review #4

    geometry assignment find the length of each arc

  6. How to Find Arc Length with Angle Equations

    geometry assignment find the length of each arc

VIDEO

  1. MTH101 Calculus & Analytical Geometry Assignment 1 Fall 2023 virtual University of Pakistan

  2. The Maths Prof: Finding the Length of an Arc

  3. ARC LENGTH

  4. Calculus 2: Arc Length Problems (Part I)

  5. How to calculate area, perimeter and lengh in ArcGis

  6. Arc Length Calculus (Example 3)

COMMENTS

  1. PDF Arc Length and Sector Area

    Find the length of each arc. Round your answers to the nearest tenth. 1) 11 ft 315 ° 60.5 ft 2) 13 ft 270 ° 61.3 ft 3) 16 ft 3 π 2 75.4 ft 4) 13 in π 6 6.8 in 5) r = 18 cm, θ = 60 ° 18.8 cm 6) r = 16 m, θ = 75 ° 20.9 m 7) r = 9 ft, θ = 7π 4 49.5 ft 8) r = 14 ft, θ = 19 π 12 69.6 ft Find the length of each arc. Do not round. 9) 8 cm ...

  2. Arc length (practice)

    Arc length. Google Classroom. A circle has a radius of 3 . An arc in this circle has a central angle of 340 ∘ . What is the length of the arc? Either enter an exact answer in terms of π or use 3.14 for π and enter your answer as a decimal. 340 ∘ 3. Learn for free about math, art, computer programming, economics, physics, chemistry ...

  3. Arc Length Calculator

    Calculate the arc length according to the formula above: L = r × θ = 15 × π/4 = 11.78 cm. Calculate the area of a sector: A = r² × θ / 2 = 15² × π/4 / 2 = 88.36 cm². You can also use the arc length calculator to find the central angle or the circle's radius. Simply input any two values into the appropriate boxes and watch it ...

  4. Radians & arc length (practice)

    Radians & arc length. Google Classroom. Write a formula for the arc length S in terms of r for the following figure. 5.6 r S. The angle in the figure is a central angle in radians. S =. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit ...

  5. Challenge problems: Arc length 1 (article)

    Now we set up a proportion to find the part of circumference intercepted by that angle, so Arc length/Circumference = Arc measure/Degrees in a circle. Arc length/16pi = 120/360. We can simplify the fraction to the right and it will be Arc length/16pi = 1/3. Since we need to isolate the arc length, we can multiply both sides by 16pi.

  6. How to Find Arc Length: Formulas and Examples

    For example, if the arc's central angle is 2.36 radians, your formula now looks like this: . 4. Multiply the radius by the arc's central angle. The product gives you the length of the arc. For example: So, the length of an arc of a circle with a radius of 10 cm and a central angle of 23.6 radians, is about 23.6 cm.

  7. PDF Length of Arc Sheet 1

    Find the arc length of each sector. Round the answer to two decimal places. ( use =3.14 ) Example: Arc length of a sector (s) = 65! x 3.14 x 7 = 7.94 ft central angle 180! x x radius = " x x r 180! = 180! Length of the arc CD 1) Length of the arc ST = 2) Length of the arc CD = 3) Length of the arc PQ = 4) Length of the arc GH = 5) Length of the ...

  8. Arc Length ( Read )

    Arcs can also be measured in length, as a portion of the circumference. Arc length is the length of an arc or a portion of a circle 's circumference. The arc length is directly related to the degree arc measure. Arc Length Formula: The length of A B ^ = m A B ^ 360 ∘ ⋅ π d or m A B ^ 360 ∘ ⋅ 2 π r.

  9. Arc Length ( Read )

    This concept teaches students to find the length of an arc. Click Create Assignment to assign this modality to your LMS. We have a new and improved read on this topic.

  10. Measures of Arcs of Circles Worksheets

    Length: The length of an arc is directly proportional to the circumference of the circle and is dependent on both the central angle and the radius of the circle. You can calculate the length of an arc is equal to the circumference times the fraction of the circle represented by the arc's measure. Length = (measure of arc/360) 2πr.

  11. Arc Length Calculator

    Now we just need to find that area. The area can be found by the formula A = πr2. Plugging our radius of 3 into the formula we get A = 9π meters squared or approximately 28.27433388 m2. Now we multiply that by 15 (or its decimal equivalent 0.2) to find our sector area, which is 5.654867 meters squared. Find the length of an arc and the area ...

  12. PDF Circumference and Arc Length Worksheet

    Circumference and Arc Length Find the circumference of each circle. Use your calculator's value of π. Round your answer to the nearest tenth. 1) radius = 6 m 2) radius = 5 yd 3) radius = 7.5 in 4) radius = 2 km 5) diameter = 8 cm 6) diameter = 19.4 mi 7) diameter = 24 km 8) diameter = 18 mi Find the length of each arc.

  13. Geometry, Circles: Area of Sectors Flashcards

    A square with sides of 3√2 is inscribed in a circle. What is the area of one of the sectors formed by the radii to the vertices of the square? 2.25 π. Study with Quizlet and memorize flashcards containing terms like How is the area of a semicircle found?, Find the length of an arc of 40° in a circle with an 8 inch radius., What is the ...

  14. Finding arc measures (video)

    One hundred eighty degrees. And so if we wanna look at this whole angle, the angle that intercepts the major arc A, B, C, is going to be 180 degrees plus 69 degrees. So we're going to have 180 degrees, plus 69 degrees which is equal to, what is that, 249, 249 degrees. That's the arc measure of this major arc A, B, C.

  15. PDF Circumference and Arc Length

    11.1 Circumference and Arc Length 583 SELF-ASSESSMENT 1 I do not understand. 2 I can do it with help. 3 I can do it on my own. 4 I can teach someone else. EXAMPLE 2 Finding and Using Arc Lengths Find each indicated measure. a. arc length of AB b. circumference of ⊙Z c. m RS P A B 60° 8 cm Z X 40° Y 4.19 in. T S R 15.28 m 44 m SOLUTION a.

  16. PDF 9.4 Circles and Arcs NAME: CORRECTIVE ASSIGNMENT DATE:

    Find the length of each arc. Round your answers to the nearest tenth. 20) 19) 16 mi 14 km Find the length of each arc. Leave your answer in terms of pi. 21) 15) 17) mCDF 120. 16) 18) Find the measure of the arc or central angle indicated. Assume that lines which appear to be diameters are actual diameters. 13) maww 14) mFEG 130.

  17. PDF 11.1 Circumference and Arc Length

    Section 11.1 Circumference and Arc Length 599 Using Arc Lengths to Find Measures Find each indicated measure. a. arc length of AB b. circumference of ⊙Z c. m RS P A B 60° 8 cm Z X 40° Y 4.19 in. T S R 15.28 m 44 m SOLUTION a. Arc length of AB 60= —° 360° ⋅ 2π(8) ≈ 8.38 cm b. Arc length of XY —— C = m XY — 360° c. Arc length ...

  18. Arc of a Circle Calculator

    Arc of a Circle Calculator. Whether you need to calculate the length of an arc, chord length, or other properties of a circle, our calculator is here to make your life easier. How to use the Calculator: Enter the diameter to calculate the radius or input the radius to find the diameter. Enter either the angle of the arc or the length of the arc.

  19. PDF Circles Arc Length and Sector Area Easy

    Geometry Assignment Name Date ID: 1 Period math-worksheet.org Find the length of each arc. Round your answers to the nearest tenth. 8 km A) 50.3 km C) 135 5 mi A) 10602.9 mi ... Find the length of each arc. Round your answers to the nearest tenth. 315 A) 7.9ft C) 603.2 ft

  20. Finding arc measures with equations (video)

    So this angle right over here has a measure of 147 degrees and you can calculate, that's the same thing as over here. 2 times -3 is -6, plus 153 is 147 degrees, these two are the same, and so 147 degrees. This angle measures the same as the measure of arc BC. Let's do one more of these. Circle P is below.

  21. Geometry B, Assignment 9. Quiz 2: Tangents, Arcs, and Chords

    If a circle is divided into 12 equal arcs, then each arc measures 30 degrees. True. The radius of a sphere is 6 inches. Find the length of a chord connecting two perpendicular radii. x = inches. 6 sqrt 2. If AXB names a major arc of a circle, then AB must be a minor arc. True. If chord AB is 3 inches from the center and chord CD is 5 inches ...

  22. Arc Length ( Read )

    Arcs can also be measured in length, as a portion of the circumference. Arc length is the length of an arc or a portion of a circle's circumference. The arc length is directly related to the degree arc measure. Arc Length Formula: If d is the diameter or r is the radius, the length of A B ^ = m A B ^ 360 ∘ ⋅ π d or m A B ^ 360 ∘ ⋅ 2 ...

  23. Geometry B, Assignment 15. Quiz 2: Area of Circles Flashcards

    A = 4 π - 8. Find the difference in area between the circle and the triangle. Click on the answer until the correct answer is showing. ( 60, 3 ) A = 9 π - 9/2 √ 3. Find the combined area of the triangles. Click on the answer until the correct answer is showing. ( 4, 90 ) A = 16.