With the first qs, why would 17pi be incorrect? All I did was find the circumference of the circle which is 60pi then divided it by 2 because the diameter indicated to me that we were only focusing on the right side of the circle. Then I subtracted 13pi from 30pi to get 17pi; and if you were to place 17pi into the missing length, you would still get a circumference of 60pi....I still don't get why this method is wrong...

You are thinking of arc length not arc measure arc length is the length of the arc arc measure is the measure of the angle not the arc This confused me as well

Why do the explanations have you find the circumference and set up a ratio when you can do the problem so much simpler? A couple videos back Sal taught that the arc measure times the radius equals the arc length. In problem 1, just divide the arc length (13pi) by the radius (30) to get the arc measure of BC (13pi/30). Then subtract that from 2pi and you get 17pi/30, which is the correct answer. Is it just me or does the whole 'finding circumference and setting up ratio' thing seem a bit unnecessary?

While I used the same method as you, it is important to know how to do these problems the long way so that you can solve problems that this shortcut doesn't work for.

Hey guys, I have a question for problem 2. According to the definition of radian, arc length divided by radius equals to arc measure(in radians) ( e.g. arc length r / radius r = arc measure 1 radian). If this is true, since angle BPD = pi/2, I can know the arc length of BCD is pi/2 * 27 = 27pi/2 ( arc measure * radius r = arc length). To find the arc length of arc length CD, I can just deduct the length of arc BC from the arc length BCD, 27pi/2 - 81pi/10 = 135pi/10 - 81pi/10 = 54pi/10 = 27pi/5, which is clearly the wrong answer for this problem. Can anyone tell me what I did wrong here? Is it the assumption that arc length divided by radius equals to arc measure(in radians) is wrong? Or is it something else? Thanks!

You appear to have the right question, but did not follow through, the arc length divided by the radius equals the arc measure. Since you got the arc length to be 27π/5, if you divide this by the radius 27, you get the arc measure which is π/5.

Can someone explain the proportion to me in number 1?

the arc LENGTH to circumference (a length) = arc measure (angle measurement) to radians in the circle (angle measurement) It is like a percentage, part divided by the whole. The parts are arc length and arc measure; the whole are circumference and radians in a circle. 2pi

Can somebody please tell me where pi/2 comes from in the question

∠𝐴𝑃𝐵 is a right angle, so its measure is 𝜋/2 radians. It's a bit unnecessary to include it in the calculations, though, because if three angles are supplementary and one of the angles is a right angle, then the other two angles are complementary, and thus: 3𝜋/10 + 𝑚∠𝐶𝑃𝐷 = 𝜋/2 ⇔ 𝑚∠𝐶𝑃𝐷 = 𝜋/5

Main content

Course: Geometry (all content) > Unit 14

Lesson 5: Arc length (from radians)

Challenge problems: Arc length (radians) 2

Google Classroom

Solve two challenging problems that ask you to find an arc measure using the arc length.

Problem 1

In the figure below,

\overset{―}{A C}

is a diameter of circle

P

. The radius of circle

P

30

units. The arc length of

\overset{⌢}{B C}

13 π

What is the arc measure of $\overset{⌢}{A B}$ ‍, in radians?

We are trying to figure out the arc measure of the arc that begins at point

A

and ends at point

B

The measure of an arc is equal to the measure of the central angle that intercepts it. So, we need to know the measure of

∠ A P B

We need to know the total circumference in order to determine the arc measure for

\overset{⌢}{B C}

$\begin{aligned} Circumference & = 2 π r \\ = 2 π (30) \\ = 60 π \end{aligned}$ ‍

We know the length of

\overset{⌢}{B C}

, so we set up a proportion to figure out its arc measure.

$\begin{aligned} \frac{arc length}{circumference} & = \frac{arc measure}{radians in a circle} \\ \frac{13 π}{60 π} & = \frac{arc measure}{2 π} \\ arc measure & = 2 π \cdot \frac{13 π}{60 π} \\ = \frac{13}{30} π \end{aligned}$ ‍

The arc measure of

\overset{⌢}{B C}

\frac{13}{30} π

. Likewise, the measure of

∠ B P C = \frac{13}{30} π

Because

\overset{―}{C A}

is a diameter,

∠ A P B

and

∠ B P C

are supplementary. So, their measures add up to

π

$\begin{array}{r} \frac{13}{30} π + m ∠ A P B = π \\ m ∠ A P B = \frac{17}{30} π \end{array}$ ‍

The measure of $\overset{⌢}{A B}$ ‍ is $\frac{17}{30} π$ ‍.

Problem 2

In the figure below,

\overset{―}{A D}

is a diameter of circle

P

. The radius of circle

P

27

units. The arc length of

\overset{⌢}{B C}

\frac{81}{10} π

What is the arc measure of $\overset{⌢}{C D}$ ‍, in radians?

We are trying to figure out the arc measure of the arc that begins at point

C

and ends at point

D

The measure of an arc is equal to the measure of the central angle that intercepts it. So, we need to know the measure of

∠ C P D

We need to know the total circumference in order to determine the arc measure for

\overset{⌢}{B C}

$\begin{aligned} Circumference & = 2 π r \\ = 2 π (27) \\ = 54 π \end{aligned}$ ‍

We know the length of

\overset{⌢}{B C}

, so we set up a proportion to figure out its arc measure.

$\begin{aligned} \frac{arc length}{circumference} & = \frac{arc measure}{radians in a circle} \\ \frac{(\frac{81}{10} π)}{54 π} & = \frac{arc measure}{2 π} \\ arc measure & = \frac{81}{10} π \cdot \frac{2 π}{54 π} \\ = \frac{3}{10} π \end{aligned}$ ‍

The arc measure of

\overset{⌢}{B C}

\frac{3}{10} π

. Likewise, the measure of

∠ B P C = \frac{3}{10} π

Because

\overset{―}{A D}

is a diameter,

∠ A P B

∠ B P C

, and

∠ C P D

are supplementary. So, their measures add up to

π

$\begin{array}{r} \frac{π}{2} + \frac{3}{10} π + m ∠ C P D = π \\ m ∠ C P D = \frac{π}{5} \end{array}$ ‍

The measure of $\overset{⌢}{C D}$ ‍ is $\frac{π}{5}$ ‍ radians.

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Natsu and Happy
Posted 8 years ago. Direct link to Natsu and Happy's post “With the first qs, why w...”
With the first qs, why would 17pi be incorrect? All I did was find the circumference of the circle which is 60pi then divided it by 2 because the diameter indicated to me that we were only focusing on the right side of the circle. Then I subtracted 13pi from 30pi to get 17pi; and if you were to place 17pi into the missing length, you would still get a circumference of 60pi....I still don't get why this method is wrong...
Button navigates to signup pageButton navigates to signup page
(15 votes)
Answer
- Samyak Jain
  Posted 8 years ago. Direct link to Samyak Jain's post “You are thinking of arc l...”
  You are thinking of arc length not arc measure
  
  arc length is the length of the arc
  arc measure is the measure of the angle not the arc
  
  This confused me as well
  Button navigates to signup page
  (16 votes)
Abbey
Posted 8 years ago. Direct link to Abbey's post “Why do the explanations h...”
Why do the explanations have you find the circumference and set up a ratio when you can do the problem so much simpler? A couple videos back Sal taught that the arc measure times the radius equals the arc length. In problem 1, just divide the arc length (13pi) by the radius (30) to get the arc measure of BC (13pi/30). Then subtract that from 2pi and you get 17pi/30, which is the correct answer.

Is it just me or does the whole 'finding circumference and setting up ratio' thing seem a bit unnecessary?
Button navigates to signup pageComment on Abbey's post “Why do the explanations h...”
(9 votes)
Answer
- C@7L0VR
  Posted 8 years ago. Direct link to C@7L0VR's post “While I used the same met...”
  While I used the same method as you, it is important to know how to do these problems the long way so that you can solve problems that this shortcut doesn't work for.
  Button navigates to signup page
  (9 votes)
Vinit Srivastava
Posted 8 years ago. Direct link to Vinit Srivastava's post “why so long? there is mor...”
why so long?
there is more simple way to do both question.for example 1st question:
arc measure =arc length/radius(13pi/30)
then pi-(13pi/30)=17pi/30
DONE! in 2 steps
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(6 votes)
Answer
- samandjam
  Posted 5 months ago. Direct link to samandjam's post “This works sometimes, but...”
  This works sometimes, but not always. If you get a question where you can't use this, you'll be stuck, so you're going to have to know this method.
  Button navigates to signup page
  (1 vote)
Wei Chen
Posted 7 years ago. Direct link to Wei Chen's post “Hey guys, I have a questi...”
Hey guys, I have a question for problem 2. According to the definition of radian, arc length divided by radius equals to arc measure(in radians) ( e.g. arc length r / radius r = arc measure 1 radian). If this is true, since angle BPD = pi/2, I can know the arc length of BCD is pi/2 * 27 = 27pi/2 ( arc measure * radius r = arc length). To find the arc length of arc length CD, I can just deduct the length of arc BC from the arc length BCD, 27pi/2 - 81pi/10 = 135pi/10 - 81pi/10 = 54pi/10 = 27pi/5, which is clearly the wrong answer for this problem. Can anyone tell me what I did wrong here? Is it the assumption that arc length divided by radius equals to arc measure(in radians) is wrong? Or is it something else? Thanks!
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(2 votes)
Answer
- David Severin
  Posted 7 years ago. Direct link to David Severin's post “You appear to have the ri...”
  You appear to have the right question, but did not follow through, the arc length divided by the radius equals the arc measure. Since you got the arc length to be 27π/5, if you divide this by the radius 27, you get the arc measure which is π/5.
  Comment on David Severin's post “You appear to have the ri...”
  (5 votes)
shaanth21
Posted 7 years ago. Direct link to shaanth21's post “Can someone explain the p...”
Can someone explain the proportion to me in number 1?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Art
  Posted 7 years ago. Direct link to Art's post “the arc LENGTH to circumf...”
  the arc LENGTH to circumference (a length) = arc measure (angle measurement) to radians in the circle (angle measurement) It is like a percentage, part divided by the whole. The parts are arc length and arc measure; the whole are circumference and radians in a circle. 2pi
  Button navigates to signup page
  (4 votes)
Uddip Kashyap
Posted a year ago. Direct link to Uddip Kashyap's post “why did you multiply by 2...”
why did you multiply by 2pi in the second problem?
Button navigates to signup pageComment on Uddip Kashyap's post “why did you multiply by 2...”
(3 votes)
Answer
- Josh Meyers
  Posted 5 months ago. Direct link to Josh Meyers's post “The only thing you have t...”
  The only thing you have to do is find the circuference of the circle and then when you try to find CD you devide the circumference by 4 since it is in the corner then you take that solution and take away 81 over 10 pie and there is your answer
  Button navigates to signup page
  (1 vote)
Nicolo-72
Posted 7 years ago. Direct link to Nicolo-72's post “Can somebody please tell ...”
Can somebody please tell me where pi/2 comes from in the question
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(2 votes)
Answer
- Jerry Nilsson
  Posted 7 years ago. Direct link to Jerry Nilsson's post “∠𝐴𝑃𝐵 is a right angle,...”
  ∠𝐴𝑃𝐵 is a right angle, so its measure is 𝜋/2 radians.
  
  It's a bit unnecessary to include it in the calculations, though, because if three angles are supplementary and one of the angles is a right angle, then the other two angles are complementary, and thus:
  3𝜋/10 + 𝑚∠𝐶𝑃𝐷 = 𝜋/2 ⇔ 𝑚∠𝐶𝑃𝐷 = 𝜋/5
  Comment on Jerry Nilsson's post “∠𝐴𝑃𝐵 is a right angle,...”
  (3 votes)
laddhanishtha
Posted 7 years ago. Direct link to laddhanishtha's post “can we use radians for ot...”
can we use radians for other figures like spheres, cylinders, etc? if so how? if not, why not? much appreciated!
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(2 votes)
Answer
- David Severin
  Posted 7 years ago. Direct link to David Severin's post “Most of what you do with ...”
  Most of what you do with spheres, cylinders, and cones is finding volume and surface area, so while you will use π, there is not a lot of need for angles and degrees/radians for 3d figures. My intuition feels there is no need to use radians fro these figures.
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  (1 vote)
Bipin Dhakal
Posted 7 years ago. Direct link to Bipin Dhakal's post “points A and B lies on a ...”
points A and B lies on a circlr with radius 1 and arc AB has length π/3. what fraction of circumferenceof the circle is the length of arc AB?
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(1 vote)
Answer
- Jayanthika
  Posted a month ago. Direct link to Jayanthika's post “The arc length 𝜋/3 / 2𝜋...”
  The arc length 𝜋/3 / 2𝜋 = 1/6 of the circle's circumference.
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  (1 vote)
Ms.Tajbibi Shamim
Posted 7 years ago. Direct link to Ms.Tajbibi Shamim's post “Will writing the arc meas...”
Will writing the arc measurement of BC wrote 13pi so now why write 17pi/30?
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(1 vote)
Answer
- alexa.pomerantz
  Posted 7 years ago. Direct link to alexa.pomerantz's post “Are you asking how to sol...”
  Are you asking how to solve the first problem?
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  (1 vote)