Find arc length given chord, radius

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Discussion Overview

The discussion revolves around calculating the arc length of a circle given the radius and the length of the chord. Participants explore the geometric relationships involved, including the properties of triangles and the use of trigonometric functions, while addressing uncertainties related to assumptions in their calculations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant expresses difficulty in finding the arc length, suggesting the answer is pi.
  • Another participant encourages drawing the scenario to aid in understanding the problem.
  • A question is raised about the type of triangle formed by the chord and the radii, prompting a discussion about the angles involved.
  • One participant reflects on the limitations of assuming certain geometric truths based on their textbook's content.
  • Another participant argues that external knowledge should not be disregarded, as it may lead to missing the intended pedagogical point.
  • A participant calculates the height of the segment and the angles involved using the Pythagorean theorem and trigonometric functions, arriving at an angle of 30° for half the central angle.
  • Further clarification is provided that the angle AOE is half the central angle, leading to a corrected angle of 60° for the full central angle.
  • One participant derives the arc length formula and confirms that the arc length is indeed pi, based on their calculations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial assumptions regarding the triangle's properties and the implications for the arc length calculation. There are competing views on the necessity of adhering strictly to the textbook's content versus applying broader mathematical principles.

Contextual Notes

Participants express uncertainty about the assumptions they can make based on their textbook, particularly regarding the properties of triangles and the angle sum theorem. The discussion includes various mathematical steps that are not fully resolved, particularly in the context of deriving the arc length.

Who May Find This Useful

This discussion may be useful for students grappling with similar geometry problems, particularly those involving arc lengths, chords, and the properties of circles.

Ragnarok7
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The solution to this question (whose answer is pi) is eluding me:

The radius of a circle is 3 feet. Find the approximate length of an arc of this circle, if the length of the chord of the arc is 3 feet also.
 
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Ragnarok said:
The solution to this question (whose answer is pi) is eluding me:

The radius of a circle is 3 feet. Find the approximate length of an arc of this circle, if the length of the chord of the arc is 3 feet also.

Hi Ragnarok! :)

Did you make a drawing?
If you draw it, that should help to find the answer...
 
Ragnarok said:
The solution to this question (whose answer is pi) is eluding me:

The radius of a circle is 3 feet. Find the approximate length of an arc of this circle, if the length of the chord of the arc is 3 feet also.

For starters, what kind of triangle made by the chord and two radii? What does that tell you about its angles?
 
Thank you! I understand now. I was unsure how much information we were allowed to assume as I can't remember if the book proved triangles have angle sum of 180 degrees yet.
 
Ragnarok said:
Thank you! I understand now. I was unsure how much information we were allowed to assume as I can't remember if the book proved triangles have angle sum of 180 degrees yet.

Just because a book hasn't shown something doesn't make it any less true or provide any reason why you can't use it.
 
True, but I usually try to stay within the internal consistency of the book because if I bring in things from outside it probably means I'm missing out on the intended pedagogical point of the exercise, and possibly missing a simpler, more clever solution.

But I am pretty sure this book presupposes a knowledge of Euclid, so I think the sum of a triangle's angles is fine to assume.
 
https://www.physicsforums.com/attachments/2013
We know radius AO (3) and chord AB.
AE = 1/2 AB
From Pythagorean Theorem OE² = AO² - AE²
OE² = 3² - 1.5²
OE² = 9 - 2.25
OE = 2.5980762114
Segment Height ED = Radius AO - Apothem OE
Segment Height ED = 3 - 2.5980762114
Segment Height ED = 0.4019237886
Angle AOE = arc tangent (AE/OE)
Angle AOE = arc tan (1.5/2.5980762114)
Angle AOE = 29.9999999996 or 30° rounded

There are 2PI radians in a circle and 30° is 1/12 of a circle.
So, 30° = 2PI/12 radians or PI/6 radians.
The answer you said was supposed to be PI. Well I get PI/6.
************************************************************
EDITED TO ADD:
Angle AOE is only half the central angle, so it should be 60° or PI/3 radians.
 
Last edited:
Let $$\frac{\theta}{2}=\angle AOE$$ then $$\theta=\angle AOB$$ and the arc length is:

$$s=r\theta=3\left(2\cdot\frac{\pi}{6} \right)=\pi$$
 

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