English
عربى
中文
Deutsch
Español
Français
हिन्दी
Hrvatski
Italiano
日本語
Português
Română
Русский
Svenska
Türkçe
Tiếng ViệtCircles and PiSphere Volume
Volume of a Sphere
To find the volume of a sphere, we once again have to use Cavalieri’s Principle. Let’s start with a hemisphere – a sphere cut in half along the equator. We also need a cylinder with the same radius and height as the hemisphere, but with an inverted cone “cut out” in the middle.
As you move the slider below, you can see the cross-section of both these shapes at a specific height above the base:
Let us try to find the cross-sectional area of both these solids, at a distance height h above the base.
The cross-section of the hemisphere is always a
The radius x of the cross-section is part of a right-angled triangle, so we can use
Now, the area of the cross section is
| A | = |
The cross-section of the cut-out cylinder is always a
The radius of the hole is h. We can find the area of the ring by subtracting the area of the hole from the area of the larger circle:
| A | = | |
| = |