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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, start overline, A, C, end overline is a diameter of circle P. The length of start overline, P, C, end overline is 25 units.
A circle that is centered around point P. Points A, B, and C are on the circle and line segment A C is a diameter. Line segments A P, B P and C P are radii of the circle that are twenty-five units long. Angle A P B is three over ten pi.
What is the exact length of B, C, A, start superscript, \frown, end superscript?

## Problem 2

In the figure below, the length of start overline, P, A, end overline is 3 units.
A circle that is centered around point P. Points A, B, C, D, and E are on the circle. Line segments A P, B P, C P, D P, and E P are radii of the circle are three units long. Angle A P B is eleven pi over eighteen. Angle C P D is pi over three. Angle C P D is a right angle.
What is the exact length of D, A, C, start superscript, \frown, end superscript on circle P?

## Problem 3

In the figure below, start overline, A, D, end overline and start overline, B, E, end overline are diameters of circle P. The length of start overline, P, B, end overline is 10 units.
A circle that is centered around point P. Points A, B, C, D, and E are on the circle. Line segments A P, B P, C P, D P, and E P are radii of the circle are ten units long. Angle B P C is pi over three. Angle A P E is nineteen pi over thirty-six.
What is the exact length of C, D, start superscript, \frown, end superscript?

## Want to join the conversation?

• 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.
• 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!
• 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.
• So arc length (in radians) is just
length = angle*radius?
• 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.
• 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.
• 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!
• The logic behind behind finding arc lengths and arc measures is related to finding the measure of the central angle that intercepts the arc. As Khan Academy solves them using proportions, it is one way but if you're not finding yourself finding comfortable with proportions as in arc length/circumference = arc measure/360, you can use other ways too.
Most questions ask you to find arc lengths and arc measures, which you can easily find by finding the measure of the central angle the arc subtends and use the 1 or 2 formulas to find arc length.
(1 vote)
• In problem 3, how does
7/36π ⋅ 20π/2π = 35/18π
?
I don't know what process is used to get from before to after.
(1 vote)
• When you multiply the fractions, you get 7*20*pi/36*2*pi*pi or 140pi/72pi^2. Divide both sides by their greatest common factor, 4pi, and you get a simplified 35/18pi. If you need more help, you can watch one of KA's videos on multiplying and simplifying fractions. Keep exploring!
• What happens if the central angle is in degrees and the radius is in feet? How do you solve it ?
(1 vote)
• It should not change anything, the examples use the generic term units, so if you said, for example, in the first problem 25 feet instead of 25 units, when you set up a proportion and solve, the answer would be in feet rather than units.