# Thread: Length of an arc

1. ## Length of an arc

The radius of a circle is 10 meters. What is the length of a 135° arc?

135/360 = 0.375 * d * pi -> 0.375 * 20 meters * 3.14 = 23.55 meters.

I'm doing this problem on an online Geometry program. 23.55 is the right answer but it will only take it in the form of 15pi/2 meters. Here's a screenshot:
http://prntscr.com/2d8pi3

How do I arrive to that simplification from 23.55?

2. ## Re: Length of an arc

23.55 is only an APPROXIMATION. The EXACT answer is \displaystyle \displaystyle \begin{align*} \frac{15\pi}{2} \end{align*}.

Just multiply \displaystyle \displaystyle \begin{align*} \frac{135}{360} \cdot 20 \end{align*}, cancel any common factors, and multiply by \displaystyle \displaystyle \begin{align*} \pi \end{align*}.

3. ## Re: Length of an arc

Originally Posted by Iceycold
The radius of a circle is 10 meters. What is the length of a 135° arc?

135/360 = 0.375 * d * pi -> 0.375 * 20 meters * 3.14 = 23.55 meters.

I'm doing this problem on an online Geometry program. 23.55 is the right answer but it will only take it in the form of 15pi/2 meters. Here's a screenshot:
http://prntscr.com/2d8pi3

How do I arrive to that simplification from 23.55?
you don't.

you recognize that 135 degrees is 3pi/4 radians and that the length of your arc is given by

$\displaystyle arclen=10 \cdot \left(\frac{3\pi}{4}\right)=\frac{30\pi}{4}=\frac{ 15\pi}{2}$

4. ## Re: Length of an arc

Originally Posted by romsek
you don't.

you recognize that 135 degrees is 3pi/4 radians and that the length of your arc is given by

$\displaystyle arclen=10 \cdot \left(\frac{3\pi}{4}\right)=\frac{30\pi}{4}=\frac{ 15\pi}{2}$
What the OP did is fine (and the OP probably has not encountered radian measurement before). It is perfectly correct to use your angle as the proportion of the circle swept out, and so the arclength is also the same proportion of the circumference. The problem was the converting to decimals (in particular, approximating \displaystyle \displaystyle \begin{align*} \pi \end{align*}).

5. ## Re: Length of an arc

Originally Posted by Prove It
23.55 is only an APPROXIMATION. The EXACT answer is \displaystyle \displaystyle \begin{align*} \frac{15\pi}{2} \end{align*}.

Just multiply \displaystyle \displaystyle \begin{align*} \frac{135}{360} \cdot 20 \end{align*}, cancel any common factors, and multiply by \displaystyle \displaystyle \begin{align*} \pi \end{align*}.
Okay, so for the next problem I got: Radius is 7 and the arc is 135 degrees.

$\displaystyle 135/360 * 14 = 5.25$

I'm stumped on what common factors to cancel out there.

6. ## Re: Length of an arc

Originally Posted by Iceycold
Okay, so for the next problem I got: Radius is 7 and the arc is 135 degrees.

$\displaystyle 135/360 * 14 = 5.25$

I'm stumped on what common factors to cancel out there.
$\displaystyle \frac{135}{360}=\frac{3}{8}$

7. ## Re: Length of an arc

I see, confused that with something else, so answer would be 3pi/8 according to post #2?

This is high school Geometry if it helps.

8. ## Re: Length of an arc

Originally Posted by Iceycold
I see, confused that with something else, so answer would be 3pi/8 according to post #2?

This is high school Geometry if it helps.
with radius 7 you have arclen =2 pi r * (135/360) = 14 pi * 135/360 = 14 * 3/8 pi = 42/8 pi = 21/4 pi

9. ## Re: Length of an arc

Originally Posted by romsek
with radius 7 you have arclen =2 pi r * (135/360) = 14 pi * 135/360 = 14 * 3/8 pi = 42/8 pi = 21/4 pi
Great, that answer went through, how did you arrive to this part? 14 * 3/8 pi = 42/8 pi = 21/4 pi

10. ## Re: Length of an arc

Originally Posted by Iceycold
Great, that answer went through, how did you arrive to this part? 14 * 3/8 pi = 42/8 pi = 21/4 pi
Multiplication is commutative. $\displaystyle 14\pi \dfrac{3}{8} = 14\cdot \dfrac{3}{8} \pi$

11. ## Re: Length of an arc

the arc-length of a circular arc is proportional to both the radius and the angle swept out.

This is what is meant by the formula:

Circumference = 2*pi*r (where r is the length of the radius).

So if the angle swept out is an entire circle, the the arc-length is 2*pi*r.

If the angle swept out is part of the circle, say n degrees, then the arc-length is:

(n/360)(2*pi*r).

For example, if we have a circle of radius 2, and the angle swept out is 180 degrees (a semi-circle), the arc-length is:

(180/360)(2*pi*2) = (1/2)(2*pi*2) = (1/2)(4*pi) = 2*pi.

This formula works for both problems given above:

(135/360)(2*pi*10) = [(3*3*3*5)/(2*2*2*3*3*5)](2*pi*10) = (3/(2*2*2))(2*pi*10) = (3/8)(20*pi) = (60/8)pi = [(4*15)/(4*2)]pi = (15/2)pi <---same answer as above

(135/360)(2*pi*7) = (3/8)(14*pi) = [(2*3*7)/(2*2*2)]pi = [(3*7)/(2*2)]pi = (21/4)pi.

Really, this is just "math with fractions". You should hopefully know the following:

1) the total circumference of a circle of radius r is 2*pi*r

2) the total angle swept out in one turn of a complete circle is 360 degrees.

So if your arc has an angle of n degrees, it is an arc that is n/360-ths of a circle, and we multiply THAT fraction times the total circumference (so 180 degrees is "half a circle" so for an arc of 180 degrees, we halve the total circumference).

For many commonly used angles (such as 30, 60, 90 for example) the fraction n/360 can be "reduced" considerably, by cancelling common prime factors from the numerator and denominator. This seems to be where you are struggling.