Sphere's in Cylinders

Question

I have an intriguing question about Spheres and cylinders that have been enabling me to understand the topic.

What is the formula for finding the volume of a sphere with the same radius and height of a cylinder(vis- versa)? I have been searching the web and still have not found a clear answer. Please help me.

Thank you!

Answer 1

you can find the volumes separately and then compare the volumes

V=(4/3)∏r^3 for the volume of a sphere

V=∏r^2h for the volume of a cylinder

if you want to know what conditions must exist in order for the volumes to be equal, set the volumes equal to each other assuming the two radii of the sphere and cylinder are equal (condition 1)

(4/3)∏r^3=∏r^2h

divide both sides by ∏r^2 to get...

(4/3)r=h

this means that if the height of the cylinder is 4/3 of the radius (of either solid), then the volumes will be equal (condition 2)

in your example, the volume of the sphere varies depending on its radius

Answer 2

When you say same radius and height, I assume you mean that the sphere sits nicely inside the cylinder (i.e. is inscribed). If the sphere has radius r, then the cylinder should have radius r and height 2r (because the "height" of the sphere is equal to its diameter, which is twice its radius). The formulas for volume now tell us:

Volume of a sphere = 4/3 × π × r³

Volume of a cylinder = (area of base) × height = π × r² × 2r = 2π × r³.

Thus the volume of the sphere is 2/3 the volume of the cylinder.