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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?
Choose 1 answer:
Choose 1 answer:

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?
Choose 1 answer:
Choose 1 answer:

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?
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • old spice man green style avatar for user Robert Dunn
    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
    (34 votes)
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  • piceratops seed style avatar for user ryan garcia
    I wish there was a little screen we can use to do the work on the computer instead of wasting paper.
    (19 votes)
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  • blobby green style avatar for user mathman
    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?
    (6 votes)
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    • aqualine seed style avatar for user Dan V
      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.
      (6 votes)
  • winston default style avatar for user MasonG
    So arc length (in radians) is just
    length = angle*radius?
    (3 votes)
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  • blobby green style avatar for user laddhanishtha
    why don't the options have 85/2 pi radians? is it a convention not to write radians or something?
    (2 votes)
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    • mr pink green style avatar for user David Severin
      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.
      (4 votes)
  • boggle yellow style avatar for user Nathan Yao
    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!
    (3 votes)
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    • male robot hal style avatar for user Shreyas
      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)
  • leafers tree style avatar for user Ethan Brock
    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)
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    • piceratops ultimate style avatar for user Joanna G
      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!
      (2 votes)
  • aqualine seed style avatar for user ratwoo3460
    What happens if the central angle is in degrees and the radius is in feet? How do you solve it ?
    (1 vote)
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  • piceratops sapling style avatar for user Chris Morrissy
    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?
    (1 vote)
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  • male robot johnny style avatar for user KATT D
    what's the difference between the arc measure and the arc length. The placement of numbers on the circle isn't clear either.
    (1 vote)
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    • leaf green style avatar for user kubleeka
      Arc measure describes what portion of the circle we're considering. If we have a quarter circle, the arc measure is the same no matter the size (radius) of the circle.

      Arc length is the actual length of the curve, as if you had measured it with a string.
      (1 vote)