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# Challenge problems: Arc length 2

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

## Problem 1

In the figure below, the radius of circle $P$ is $10$ units. The arc length of $\stackrel{⌢}{ABC}$ is $16\pi$.
What is the arc measure of $\stackrel{⌢}{AC}$, in degrees?

## Problem 2

In the figure below, the radius of circle $P$ is $18$ units. The arc length of $\stackrel{⌢}{BA}$ is $14\pi$.
What is the arc measure of $\stackrel{⌢}{BC}$, in degrees?
${}^{\circ }$

## Want to join the conversation?

• In problem 2, why would arcAC + arcCB = arcAB? If the arc were labeled arcACB, I would understand, but the designation arcAB suggests that it refers to the arc on the other side of the circle, so that arcAC, arcAB and arcCB are three separate arcs that together form the full circle. There is no statement that the figure is drawn to scale, so there is no reason to assume that there are not three angles with measures 76º, 216º and 144º adding up to the full 360º of the circle.
• This threw me off as well. I was trying to add 140 and 76 and subtract that from 360... confusing. How would you refer to the large AB arc?
• in the 2nd question, why isn't arc AB called ACB for clarity?
• I agree, this problem is very tricky. I reviewed several times and realized most of my confusion was getting the distance between AB and BA confused. (AB is the longer of the two) What helped me understand was that the distance was given as 18PI. This made sense when I figured out the circumference of the whole circle was 36PI.
The distance of the section they are asking for is the smaller of the two (BCA) vs BA
After I figured that out, the hints make sense.
• well, I have a question in problem number 2... after we get 140, why did we subtract it from 76! it doesn't make sense at all! the two angles are not equal to the other (larger) and when we add 76,140,and 64, it gives us 280 not 360.. this is driving me crazy!! I spent 2 hours trying to understand this problem..
• BC is 64 degrees. AC is 76 degrees. They both add up to 140 degrees. I think what you did was double that and likely got confused in the shown work.
Conveniently in this problem, every pi length is 10 degrees, as 36pi is a full circle.
So we have 14pi, which is the length of AB. that means, the diameter created by B going through the center to A is 4pi, which makes one side with the length 18pi. To get the full circle, we double that, which gets us to 36pi.
Hopefully this helps!
• There is a lot of confusion in the comment section and I am baffled at what all of you are saying. Where did you get all these other numbers like "144" from? Can someone please explain to me in a lucid manner?
• Some people thought the required arc was major arc AB because minor arc AB has a point C.
• I do not understand that how does, the angle degrees add to 360? 140 degrees+76degrees+64degree = 280 degrees, not 360degrees as a full circle?
• Hi Jukka, I'm afraid you understood the explanation wrong. The 140 degrees is the combination of the 76 and 64 degree angle. If you wanted to check if the angles add up to 360, you would either add 76 and 64 and the major arc measure of BC or add 140 and the major arc BC. Of course, you would need another point on the circle to properly use the term 'major arc'. The problem explanation does not go into solving for the obtuse angle in the diagram. You could solve for it though: 360-140=220 degrees.
I hope that cleared any confusion, and feel free to ask if you have any more questions!
All in all, great question!
~Hannah
• Q2.
AC+CB=AB?? So all that theory about circles adding up to 360 degrees is wrong?
• Minor arc AB is just a portion of the complete circle P. The whole circle together still is 360 degrees.

Hope this helps :)
• Label in problem 2 is confusing. Better say 14pi is the arclength of ACB rather than AB.
• that is not the way we label minor arcs, you only use the two endpoints, This is to distinguish it from a major arc which does require 3 letters, I do not understand your confusion,
• mr the problem 2 from arc length 2 assignment the answer is wrong. the answer is 144 degree
• With a radius of 18, the circumference is C=2πr=36π. So the arc is 14π/36π of the circumference or 7/18 of a circle. In degrees, 7/18*360 = 140 degrees for arc BA, and since you already know 76 degrees, 140-76=64 degrees. How are you getting 144 degrees?
• In the figure above, the circle has center O and angle ∠XOZ = 40°. If the length of arc Arc XZ (shown in bold) is between 3 and 5, what is one possible integer value for the length of the radius?
• We have the arc length (at least a range of values for the arc length) and the angle, and we want to find the radius. To do this, we can set up our usual proportion, where the fraction of 360 degrees that is the arc's angle is the same as the fraction of the circumference that is the arc length:
40 / 360 = arc length / (2pi*r)
We know the arc length is between 3 and 5, so the first thing that comes to mind is to test both 3 and 5, and see what our possible values of radius come out to be:
40/360 = 3/(2pi*r)
1/9 = 3/2pi*r
2pi*r = 27
r = 27 / 2pi = 4.29...
40/360 = 5 / (2pi*r)
1/9 = 5 / 2pi*r
2pi*r = 45
r = 45 / 2pi = 7.16...
So, if we know that the length is between 3 and 5 for an arc that stretches over 40 degrees, the radius (integer values) has to be between 5 and 7, inclusive. Does this help?