If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

High school geometry

Course: High school geometry>Unit 8

Lesson 5: Arc length (from radians)

Challenge problems: Arc length (radians) 2

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

Problem 1

In the figure below, $\stackrel{―}{AC}$ is a diameter of circle $P$. The radius of circle $P$ is $30$ units. The arc length of $\stackrel{⌢}{BC}$ is $13\pi$.
What is the arc measure of $\stackrel{⌢}{AB}$, in radians?

Problem 2

In the figure below, $\stackrel{―}{AD}$ is a diameter of circle $P$. The radius of circle $P$ is $27$ units. The arc length of $\stackrel{⌢}{BC}$ is $\frac{81}{10}\pi$.
What is the arc measure of $\stackrel{⌢}{CD}$, in radians?

Want to join the conversation?

• 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.
• why so long?
there is more simple way to do both question.for example 1st question:
then pi-(13pi/30)=17pi/30
DONE! in 2 steps
• 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!
(1 vote)
• 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.
• why did you multiply by 2pi in the second problem?
• 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