10.03 Volume of spheres
Introduction
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.
Ideas
Volume of spheres
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 hypothesis 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.
How many cones of water did it take to fill the sphere?
Recall that the volume of a cone can be found using the formula V=\dfrac{1}{3}\pi r^2h. Using formula for the volume of a cone and your answer to the question above, develop 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.
The volume of the sphere is 4 times the volume of a cone with the same radius and a height that is equal to the radius.
| \displaystyle \text{Volume of sphere} | \displaystyle = | \displaystyle 4 \times \text{Volume of cone} | Formulate the equation |
| \displaystyle = | \displaystyle 4 \times \frac{1}{3}\pi r^2h | Substitute the formula | |
| \displaystyle = | \displaystyle 4 \times \frac{1}{3}\pi r^2r | Substitute h=r | |
| \displaystyle = | \displaystyle \frac{4}{3}\pi r^3 | Simplify |
Examples
Example 1
Find the volume of the sphere shown. Round your answer to two decimal places.
The volume, V, of a sphere can be calculated using the formula:
V=\frac{4}{3}\pi r^3 where r is the radius of the sphere.