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Trigonometry Find arc length given chord, radius

Ragnarok

Member
Jul 10, 2013
41
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.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
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

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
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?
 

Ragnarok

Member
Jul 10, 2013
41
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

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
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.
 

Ragnarok

Member
Jul 10, 2013
41
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.
 

wolf

New member
Feb 25, 2014
1
View attachment 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:

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Let \(\displaystyle \frac{\theta}{2}=\angle AOE\) then \(\displaystyle \theta=\angle AOB\) and the arc length is:

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