Question

How do I calculate arc length? Imagine a slice of pizza, I need to calculate the area of the slice, the interior angle, and the arc length of the slice. How would I start it? (I have to repeat this 7 more times, just want to know how to get started)

Answer 1

It's actually really easy! The formula for arclength if you have the function is:
\[\int\limits_{}^{} \sqrt{1+\frac{ dy }{ dx }}dx\] or if you have the parametric equation then it is:
\[\int\limits\limits_{}^{} \sqrt{\frac{ dx ^{2} }{ dt }+\frac{ dy ^{2} }{ dt }}dt\]

Answer 2

Thank you! :D If I could though, I need a bit more help deciphering the formula you gave me. What does dx, dt, and the symbol next to the square root sign mean?

Answer 3

Take the integral of the squareroot with respect to x or respect to t with the second formula

Answer 4

Integral?

Answer 5

This is highschool Geometry by the way.

Answer 6

This is calculus 3? Multivariable calculus?

Answer 7

My vocabulary may be big but it isn't THAT big haha.

Answer 8

Calculus 3? I had no idea that even existed xD. I doing... ehh "somewhat" simple Geometry

Answer 9

I hope I'm not wasting your time by the way x'c

Answer 10

Ahh! Ok I thought this was calc 3! The equation for arc length if you want to do it the basic way is: (2pi*r)*(n degrees/360)

But if you want to impress your teacher you could do some multivariable calculus to solve the equation!

Answer 11

Okay so I have the formula written down for arc length. For the interior angles though, I've always been rusty with that kind of stuff. Would I determine the interior angles by the arc length?

Answer 12

It makes sense if that's how

Answer 13

If you have the radius and arc length then you could find out the inner angle!

Answer 14

:D yay! Well check it out: (bare with me, I gotta type it lol)

Answer 15

I have a pizza that is 20 inches in diameter. I'm going to cut it into 8 slices, for the first step I have to find the interior angles and arc lengths of EACH slice. Since they are going to be congruent I only have to do it for ONE, I already have the tools to do that (My many thanks). I'm going to plug in what you told me and solve the first step. How long are you going to be on?

Answer 16

also, "n" is that number of degrees or angles amongst the 8 slices?

Answer 17

We know that a circle is 360 degrees and there are 8 slices so each slice is (8/360) degrees! The radius is 10 because 20/2 is 10. Just plug into the formula to get the answer!

Answer 18

Okie dokie, stand by.

Answer 19

I got 1.395 repeating

Answer 20

So my arc length of EACH slice will be 1.4? or 2?

Answer 21

Yep! It would be 1.396 or (4/9)pi

Answer 22

yesss! okay now I gotta do the AREA of the slice

Answer 23

I imagine that's going to have something to do with one-eighth and Cirumference, along with pi haha

Answer 24

The area is different than circumference!
The area is Pi r^2, but the circumference is the derivative: 2Pi*r

Answer 25

Hey I have to go eat dinner, I am extremely grateful your helping me but I haven't seen my dad in 4 months so would you be willing to resume this innnnn like 15-20 minutes? Your working magic right now and I still need your help :P lol

Answer 26

No problem!

Answer 27

:D yay! Alright, I'll be back in 15!

Answer 28

Aye you there?

Answer 29

Love it lol, reason for it?

Answer 30

no reason. What's the problem?

Answer 31

lol nothing haha.

Answer 32

my dad said that instead of 8/360 it would be 360/8. for the arc formula

Answer 33

? Hello

Answer 34

You divide the total degrees of the circle (360) by the number of pieces which is 8. So yep he's right I wrote it out wrong. Once you have that, then you plug it into n

Answer 35

ohhhh

Answer 36

Lemme redo it then.

Answer 37

Sorry about that!

Answer 38

It's all good! :D hold on

Answer 39

I got 2826

Answer 40

It would be 2.5 Pi or 7.853

Answer 41

First d 360/8 to get the inner angle, then plug that into n and solve!

Answer 42

so 360/8= n?

Answer 43

Yep!

Answer 44

okay hold on

Answer 45

okay so now i have 62.8 x 360/45. correct?

Answer 46

What??? Where did these numbers come from?! Most teachers would rather have you write the symbol for pi instead of multiplying it out

Answer 47

D: im so confused doe. I need verbal exchange for this. I'm sorry :/ your help has been appreciated though! You get medals lol. Ill jsut have my dad help me. Im tired anyways lol

Answer 48

Hope all works out!

Answer 49

Thank you! :D