Question

Why does the length of arc formula work?

Here is a picture of the formula: http://0.tqn.com/d/math/1/0/P/e/Length-of-Arc.gif

How do you use it?

Answer 1

Kiwi, do you have a basic understanding of Circumference of a circle?\[\Large\rm C=2\pi r\]

Answer 2

yes

Answer 3

Circumference ~ `Distance around the circle`

So if we want the distance `half way around` the circle.
We only want 180 of those 360 degrees, yes?

Answer 4

Correct

Answer 5

180/360 simplies to 1/2.
That would give us the length of the arc (or distance) half way around the circle.

Answer 6

For half circle
\[\Large\rm Length=\frac{1}{2}2\pi r\]

Answer 7

The weird looking fraction in front n/360 is just setting up a "portion" of the circle.

Answer 8

did you put 1/2 instead of 180/360 to simplify?

Answer 9

Do you multiply across next?

Answer 10

Ah sorry I ran away for a sec >.<
So in the example:
If we wanted to find the arc length of half of a circle,\[\Large\rm Length=\frac{180^o}{360^o}\cdot 2\pi r\]We can set it up this way because 180 degrees represents half the angle measure of a full circle.
The units cancel out since they're both degrees,\[\Large\rm Length=\frac{180^{\cancel o}}{360^{\cancel o}}\cdot 2\pi r\]Giving us,\[\Large\rm Length=\frac{180}{360}\cdot 2\pi r\]180/360 is a fraction that can be simplified,\[\Large\rm Length=\frac{1}{2}\cdot 2 \pi r\]That would give us the arc length of half of a circle with radius r.

Answer 11

Multiply across the 1/2 and 2? yes you could do that to further simplify.\[\Large\rm Length=\frac{1}{2}\cdot \frac{2}{1} \pi r\]The 2 in the numerator and 2 in the denominator can cancel out,\[\Large\rm Length=\pi r\]

Answer 12

The weird thing in front is just the uhh... portion of the total angle of the circle.
So when we did a half circle, the angle portion simplified to 1/2

Answer 13

Do you have a specific problem that you need to apply the formula to?

Answer 14

Why do you multiply straight across instead of of diagonally? Unfortunately I dont have a picture of the problem but there is a circle with an arc of 8 units. That arc has a central angle of 140 degrees. What is the radius?

Answer 15

Something like that?

Answer 16

yes

Answer 17

You only multiply diagonally `across an equality sign`.
Example:\[\Large\rm \frac{1}{2}=\frac{x}{3}\]Cross multiplying gives us,\[\Large\rm 3=2x\]But when there is no equality sign, we don't cross multiply.
We follow the normal rule for fractions, top multiply top, bottom with bottom,\[\Large\rm \frac{1}{2}\cdot\frac{x}{3}=\frac{x}{6}\]

Answer 18

That second example was alil sloppy.. I meant to say,\[\Large\rm \frac{1}{2}\cdot \frac{x}{3}\]No equality sign right?
So multiplying across,\[\Large\rm \frac{1\cdot x}{2\cdot 3}\]Which simplifies to,\[\Large\rm \frac{x}{6}\]

Answer 19

So for your problem, use the formula:\[\Large\rm Length=\frac{n}{360}2 \pi r\]They told us that,
\[\Large\rm n=140, \qquad Length=8\]

Answer 20

Plugging in the pieces,\[\Large\rm 8=\frac{140}{360}2\pi r\]Then a little math is required to solve for r.

Answer 21

You can write any whole quantity over 1 and it's the same thing, right?
8/1 is the same as 8.

So let's do that on the right side of our equation.\[\Large\rm 8=\frac{140}{360}\cdot \frac{2\pi r}{1}\]

Answer 22

We'll follow the normal rule for multiplying fractions, top with top, bottom with bottom,\[\Large\rm 8=\frac{280\pi r}{360}\]I multiplied the 2 and 140 together,
and then the 360 and 1 together.

Answer 23

Seems like you're more comfortable with cross multiplication...
So you could write your left side as 8/1 and then cross multiply from there,\[\Large\rm \frac{8}{1}=\frac{280 \pi r }{360}\]Cross multiplying gives:\[\Large\rm 2880=280 \pi r\]

Answer 24

We have a 280 and pi multiplying our r,
so to isolate the r we need to perform the inverse to get them on the other side.
We'll divide both sides by 280 and pi.\[\Large\rm \frac{2880}{280\pi}=\frac{280\pi r}{280\pi}\]They divide evenly, or cancel on the right side,\[\Large\rm \frac{2880}{280\pi}=\frac{\cancel{280\pi} ~r}{\cancel{280\pi}}\]Giving us,\[\Large\rm \frac{2880}{280\pi}=r\]

Answer 25

Which can be simplified a little bit,
or you can throw it into a calculator if you want a nice approximation.

Answer 26

What do you think?
Too crazy? Math no fun anymore?

Answer 27

What I got is 3.274044544. Wow thanks for such a detailed explanation. One more question. Why is it that you only cross multiply when there is an equality sign? Is it because we are not trying to find a variable?

Answer 28

Cross multiply is really just a silly trick.
It's not a real ... math.. thing.

Example:\[\Large\rm \frac{1}{2}=\frac{x}{3}\]If we multiply each side by 2 (I'm going to write it as 2/1),\[\Large\rm \frac{2}{1}\cdot\frac{1}{2}=\frac{x}{3}\cdot \frac{2}{1}\]simplifying:\[\Large\rm \frac{2}{2}=\frac{2x}{3}\]And simplifying further,\[\Large\rm 1=\frac{2x}{3}\]Multiplying each side by 3,\[\Large\rm 1\cdot\frac{3}{1}=\frac{2x}{3}\cdot\frac{3}{1}\]Simplfies to,\[\Large\rm 3=2x\]
Cross multiplying is all of those steps bundled up into one.
But you can see throughout the whole process I was applying the normal rule for multiplying fractions, numerators multiply together,
denominators together.

Answer 29

Thanks a lot you really helped me out. :) Yeeeesss!