MHB Find arc length given chord, radius

[Ragnarok7]

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.

 

[I like Serena]

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...

 

[Prove It]

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?

 

[Ragnarok7]

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.

 

[Prove It]

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.

 

[Ragnarok7]

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.

 

[wolf1]

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.
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EDITED TO ADD:
Angle AOE is only half the central angle, so it should be 60° or PI/3 radians.

 

[MarkFL]

Let $$\frac{\theta}{2}=\angle AOE$$ then $$\theta=\angle AOB$$ and the arc length is:

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