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# Challenge problems: Arc length (radians) 1

Solve three challenging problems that ask you to find arc length without directly giving you the arc measure.

## Problem 1

In the figure below, $\stackrel{―}{AC}$ is a diameter of circle $P$. The length of $\stackrel{―}{PC}$ is $25$ units.
What is the exact length of $\stackrel{⌢}{BCA}$?

## Problem 2

In the figure below, the length of $\stackrel{―}{PA}$ is $3$ units.
What is the exact length of $\stackrel{⌢}{DAC}$ on circle $P$?

## Problem 3

In the figure below, $\stackrel{―}{AD}$ and $\stackrel{―}{BE}$ are diameters of circle $P$. The length of $\stackrel{―}{PB}$ is $10$ units.
What is the exact length of $\stackrel{⌢}{CD}$?

## Want to join the conversation?

• I wish there was a little screen we can use to do the work on the computer instead of wasting paper.
• You can use the Math Input Panel app that is already installed on Windows, if your device is running a Windows OS (Vista/7/8/8.1/10). Hope this helps!
• Isn't it just quicker to solve the above problems by finding the arc length first by Arc = radius x angle?
Then subtract this amount from 2 Pi then multiply the result by the radius?

Thanks
Best regards
• You cannot find the arc length of CD until you have established a measure for angle CPD or a combined measure of all other angles.
• Is there are way to tell that CD (in question 3) was asking for DC in the clockwise direction, rather than CD in the anti-clockwise direction? I got my answer (correct) only by looking at the answers provided. Is there a convention that says, 'looking for angles is always clockwise'? Or did I miss something such as CD clockwise would be written as CAD,or CBAED, or something?
• When we write an arc with only two letters it implies that it is a minor arc (under 180°) so in this case CD is the arc that goes anti-clockwise from C to D, in other words the small arc.
And yes you are right about the clockwise notation : we typically use a third point to write major arcs (over 180°) in order to avoid any confusion, so CD clockwise could be CAD, CBE and so on.
• I'm so confused about how to find the lengths and the videos just aren't helping me.
• To find the length of an arc, you can apply the formula 2πr * θ / (2π)
-> r * θ.
2πr is the circle circumference.
θ / (2π) is the ratio of the angle of the arc to the whole circle (360 degree = 2π radian)
• why don't the options have 85/2 pi radians? is it a convention not to write radians or something?
• radians is a measure of angles, so if you were measuring a length, you would not use radians, the answer would be in "units" not radians. When you divide the arc measure/radians in a circle, both units are radians, and they cancel, so when you multiply by the circumference in units, then the answer is in units.
• So arc length (in radians) is just
• That is correct! For example, the arc length of an angle of 2/5 radians in a circle with a radius of 5 units is 2 units.
• The end part of changing 7pi/36 to 35pi/18,
All i did was multiply 7pi/36 by 10, the radius, then simplify.
Is this an incorrect form/doesn't work in every situation?
• This is valid, the arc length will always be the subtended angle multiplied by the radius.
• Can somebody explain the logic behind the way of solving all of these questions, I feel like I am just memorizing some formulas , thank you!