8.05 Volume of spheres

Now that we have seen how to find the volume of a cone, let's see if we can relate that to the volume of a sphere.

Exploration

Imagine a sphere that has the same radius as a cone with equal radius and height. How many times greater is the volume of the sphere than the volume of the cone?

Test your conjecture with the applet below. Click the button to pour the water from the cone to the sphere. Then, refill the water in the cone and repeat until the sphere is full.

Then, consider the following:

1. How many cones of water did it take to fill the sphere?
2. Recall that the volume of a cone can be found using the formula $V=\frac{1}{3}\pi r^2h$V=13​πr2h.  Using formula for the volume of a cone and your answer to the question above, derive the equation for the volume of a sphere.

Notice that if a cone and a sphere have equal radius, and if the cone has a height equal to its radius, then the cone will fill the sphere exactly four times. This means that the volume of the sphere is $4$4 times the volume of the cone.

$\text{Volume of sphere}$Volume of sphere	$=$=	$4\times\text{Volme of cone}$4×Volme of cone
	$=$=	$4\times\frac{1}{3}\pi r^2h$4×13​πr2h	Substitute the formula for the volume of a cone
	$=$=	$4\times\frac{1}{3}\pi r^2r$4×13​πr2r	In this special case, $h=r$h=r for the cone.
	$=$=	$\frac{4}{3}\pi r^3$43​πr3	Simplify

 

 

Practice questions

Question 1

Question 2

Question 3