# TrigonometryFind arc length given chord, radius

#### Ragnarok

##### Member
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
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
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
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
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
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
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.
The answer you said was supposed to be PI. Well I get PI/6.
************************************************************
Let $$\displaystyle \frac{\theta}{2}=\angle AOE$$ then $$\displaystyle \theta=\angle AOB$$ and the arc length is:
$$\displaystyle s=r\theta=3\left(2\cdot\frac{\pi}{6} \right)=\pi$$