13.3 - Volume of a Sphere!

Volume Formula of a Sphere:In 3 dimensions, the volume inside a sphere is found by

using the formula

where r is the radius of the sphere. Once you have the radius, you can easily find Volume!

Archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. 

• It’s important to notethat the sphere and the cylinder have the sameradius and the same height/diameter!

Math History:

I’m curious… Let’s see how this works!

Vcylinder = (πr²)•HVcylinder = (πr²)•2rVcylinder = 2πr³

Vsphere = 2/3 • Vcylinder

Vsphere = 2/3 • (2πr³)Vsphere = 4/3 • (πr³)

Let’s watch how this works!

http://www.youtube.com/watch?v=h4j8l3p22e8

Fun Facts!• A spherical triangle is a figure formed on the surface of a sphere by

three great circular arcs intersecting in three vertices. • The spherical triangle is sometimes called an Euler (oiler) triangle .• The sum of the angles of a spherical triangle are between 180°-

540°.

For a given surface area, the sphere is the one solid that has the greatest volume. This why it appears in nature so much, such as water drops, bubbles and planets.

M.C. EscherSelf-Portrait

A group of students from the Charles W Davidson College Of Engineering at San Jose State University worked on an omnidirectional, self-balancing SDS electric motorcycle. [September 14, 2012]

The SDS concept vehicle is described as a self-balancing electric motorcycle that rides on spheres. Rather than using the kind of mechanical gyros that help keep the gyrobike in an upright position, the SDS creation uses data from MEMS gyroscopic sensor technology and an onboard accelerometer to electronically control balance.

Example #1:Find the exact volume of a sphere that has a

diameter of 18 cm

Example #2:Find the exact volume of the sphere…

Example #3:Find the exact volume of the hemisphere…

Example #4:If the exact volume of a sphere is 36π cubic

inches (in³), find its diameter and its radius.