How to work out the radius of a slice of a sphere or spherical cap?

[humblemumble]

Hi there! I started making a measuring cup for 3d printing and I wanted to make really accurate measure markers on the side. I thought I'd keep it simple and make the cup a hemisphere so I could easily work out the volumes... I'm literally completely stuck on half a cup haha

How do you work out mathematically where to put a half way marker on a hemispherical cup? I have the radius and volume of the sphere, so I know the volume of the half cup. I've been looking for formulas online and found this page:

Spherical Cap -- from Wolfram MathWorld

A spherical cap is the region of a sphere which lies above (or below) a given plane. If the plane passes through the center of the sphere, the cap is a called a hemisphere, and if the cap is cut by a second plane, the spherical frustum is called a spherical segment. However, Harris and Stocker...

mathworld.wolfram.com

But this tells you how to work out the volume, and cant figure out how to rearrange the formula. Could do with a hand!

Thanks

 

[Dr.Peterson]

Hi there! I started making a measuring cup for 3d printing and I wanted to make really accurate measure markers on the side. I thought I'd keep it simple and make the cup a hemisphere so I could easily work out the volumes... I'm literally completely stuck on half a cup haha

How do you work out mathematically where to put a half way marker on a hemispherical cup? I have the radius and volume of the sphere, so I know the volume of the half cup. I've been looking for formulas online and found this page:

Spherical Cap -- from Wolfram MathWorld

A spherical cap is the region of a sphere which lies above (or below) a given plane. If the plane passes through the center of the sphere, the cap is a called a hemisphere, and if the cap is cut by a second plane, the spherical frustum is called a spherical segment. However, Harris and Stocker...

mathworld.wolfram.com

But this tells you how to work out the volume, and cant figure out how to rearrange the formula. Could do with a hand!

Which of the formulas on that page are you starting with? Some are easier than others for your purpose.

 

[humblemumble]

I was looking at this one

But if I rearrange it I need to know the value of h to solve it right?

 

[Dr.Peterson]

I was looking at this one

But if I rearrange it I need to know the value of h to solve it right?

That's the appropriate formula to use. But you know R, and you know V, so you're solving for h, right?

The trouble is that this is a cubic equation in h, which is hard to solve algebraically. I'd usually use some sort of numerical procedure (such as a graphing calculator or a similar tool can do).

 

[humblemumble]

Yes thats right i"m sovling for h, i didnt see it like that for some reason will give that a try, thank you!

A numerical procedure sounds great although i dont really know what that means?