5.+Calculate+the+arc+length+given+a+central+angle+in+radians


 * Obective #5**

__Calculating the arc length given a central angle in radians__
By Emma Lanctot

Click here for a helpful website with additional information!

Click here for a helpful step by step instructional video!!! ^^Instructional video will hold a step by step of each example! :)

I chose to create a Wikispace for this objective due to the fact that in the beginning I found it mind-bogglingly difficult. When this was brought up in class I still could not even comprehend exactly how to work with radians. However, with the introduction of this objective into the classroom I was able to connect something I had first learned in Geometry to the radians that I did not understand at the time. By creating this bridge it allowed me to understand the connection between radians and degrees.

In order to calculate the arc length of a given central angle in radians one most understand the general idea of what radians are. To find out more information about radians, check out objective 2 on this wikispace.

The next bit of information one needs is the arc length formula which is as follows: S=2πr(θ/2π) This formula holds the three variables of s, r, and θ. S stands for the arc length in whatever units the radius is given in, r is the radius of the circle, and θ is the measured central angle in radians. Now where does this formula come from? Circumference: c=2πr Think of this calculation as a portion of a circumference, using the circumference and then multiplying it by the proportion using the given central angle. 2π, being the total degrees in a circle is the denominator in the proportion. Therefore you end up with the equation s=2πr(θ/2π).


 * __EXAMPLES__**

//Example A// Known Information: Θ = π/3 Radius= 9 inches Unknown variable is S (arc length) Solve for S in inches Use the equation: S=2πr(θ/2π) to solve for S   1) Plug in known information 2πr(θ/2π)   2) 2π(9)[(π/3)/2π)]   3) 18π * (1/6)   4) =9.4248inches    Arc Length is 9.4248 inches!

// Example B // • Known information:  – S=31.27 centimeters   – R= 7 centimeters Solve for Θ in radians

Use the equation: S=2πr(θ/2π) 1) 31.27=2π(7)(θ/2π)   2) 31.27= 43.982*(θ/2π) 3) .71429= θ/2π   4) 4.488= θ Degrees in radians is 4.488

__ Portfolio Problem: __
Lauren is an avid gardener. She's looking to build a circular fence around her garden. The only thing she needs is a door that's three feet wide so she can fit her wheelbarrow through. If the garden has a diameter of nine feet, how much fence will she need, and how big will the central angle of the curved door be?

Sources: http://www.mathmistakes.info/facts/TrigFacts/learn/images/ucar.gif http://www.mathwarehouse.com/trigonometry/radians/images/picture-s%3Dr-theta-circle.gif http://www.themathpage.com/atrig/arc-length.htm