# Surface of a spherical cap

1. Jul 24, 2014

### bobie

from wiki:
Suppose we have a hemisphere of radius 10 r10 (a) and cut it in ten horizontal slices (1 is on the top), does that mean that all slices have the same surface ?

even slice 1 has surface 62.8 (2\pi *10*1)? and its a (r1) =4.36?
so, the area of slice 4 (like all others) is
2pi*10*4-2pi*10*3 = 2pi*10= 62.8

is this correct?
If it is not, what is the formula to find the area and a (r1) of slice 1?

Thanks

Last edited: Jul 24, 2014
2. Jul 24, 2014

### Simon Bridge

It is possible to cut the horizontal slices so that each has the same surface area by varying the spacing.
To work out the surface area of each slice - use calculus.

3. Jul 24, 2014

### bobie

Hi Simon, thanks. I am just starting to learn calculus.
If I understood what you said, if we cut 10 equal slices of 1 cm , they will not have the same surface?
Could you show me how to frame the equation(s)?
Thanks

4. Jul 24, 2014

### Simon Bridge

That's right - I would be surprised if the areas came out the same.

If we say that the floor is the x-y plane and up is the +z axis, then you start by dividing the whole hemisphere (radius R) into very thin disks - thickness "dz". Then you want to work out the equation for the surface area "dS" of the disk between z and z+dz in terms of z and R.

5. Jul 24, 2014

### bobie

Then I misinterpreted wiki?

Because if we find the area of the slice on the 'floor' S 9 subtracting the cap with h = 9 (2pi*10*9) = 565.48 from the hemisphere 628,3 we get 62.8
and the same happens all the way to the top to S1
Wher did I go wrong?

6. Jul 24, 2014

### Simon Bridge

Hah - I just tried it out and I am surprised ;) - try of for 2 slices.
I still think your best proof involves doing the calculus.

7. Jul 24, 2014

### bobie

I was, too, that's why I checked here, it seems amazing, right!
If you are intrigued, check by calculus, and let me know!

8. Jul 24, 2014

### MrAnchovy

9. Jul 24, 2014

### Simon Bridge

Yes. This is one of the reasons I like to answer questions here - sometimes someone surprises me.
This is the sort of thing that is obvious in retrospect.

10. Jul 24, 2014