http://www.ck12.org Chapter 11. Surface Area and Volume
To find the volume of the sphere, you need to add up the volumes of an infinite number of infinitely small pyramids.
V(all pyramids) =V 1 +V 2 +V 3 +...+Vn=1
3
(B 1 h+B 2 h+B 3 h+...+Bnh)=1
3
h(B 1 +B 2 +B 3 +...+Bn)The sum of all of the bases of the pyramids is the surface area of the sphere. Since you know that the surface area of
the sphere is 4πr^2 , you can substitute this quantity into the equation above.
=
1
3
h(
4 πr^2)
In the picture above, we can see that the height of each pyramid is the radius, soh=r.
=
4
3
r(πr^2 )=4
3
πr^3To see an animation of the volume of a sphere, see http://www.rkm.com.au/ANIMATIONS/animation-Sphere-Vo
lume-Derivation.html by Russell Knightley. It is a slightly different interpretation than our derivation.
Volume of a Sphere:If a sphere has a radiusr, then the volume of a sphere isV=^43 πr^3.
Example A
The circumference of a sphere is 26πf eet. What is the radius of the sphere?
The circumference is referring to the circumference of a great circle. UseC= 2 πr.
2 πr= 26 π
r= 13 f t.Example B
Find the surface area of a sphere with a radius of 14 feet.
Use the formula,r= 14 f t.
SA= 4 π( 14 )^2
= 784 πf t^2