There are many applications of finding the volume of a sphere. Below are just a few examples:
The volume of a sphere with radius $r$r can be calculated using the following formula:
$\text{Volume of sphere }=\frac{4}{3}\pi r^3$Volume of sphere =43πr3
We won't be expected to recreate this proof, but it can be interesting to read through.
If four points on the surface of a sphere are joined to the center of the sphere, then a pyramid of perpendicular height r is formed, as shown in the diagram. Consider the solid sphere to be built with a large number of these solid pyramids that have a very small base which represents a small portion of the surface area of a sphere.
If $A_1$A1, $A_2$A2, $A_3$A3, $A_4$A4, .... , $A_n$An represent the base areas (of all the pyramids) on the surface of a sphere (and these bases completely cover the surface area of the sphere), then,
$\text{Volume of sphere }$Volume of sphere | $=$= | $\text{Sum of all the volumes of all the pyramids }$Sum of all the volumes of all the pyramids |
$V$V | $=$= | $\frac{1}{3}A_1r+\frac{1}{3}A_2r+\frac{1}{3}A_3r+\frac{1}{3}A_4r$13A1r+13A2r+13A3r+13A4r $\text{+ ... +}$+ ... + $\frac{1}{3}A_nr$13Anr |
$=$= | $\frac{1}{3}$13 $($( $A_1+A_2+A_3+A_4$A1+A2+A3+A4 $\text{+ ... +}$+ ... + $A_n$An $)$) $r$r | |
$=$= | $\frac{1}{3}\left(\text{Surface area of the sphere }\right)r$13(Surface area of the sphere )r | |
$=$= | $\frac{1}{3}\times4\pi r^2\times r$13×4πr2×r | |
$=$= | $\frac{4}{3}\pi r^3$43πr3 |
where $r$r is the radius of the sphere.
Find the volume of a marble with a diameter of $2.3$2.3 cm, to two decimal places.
Think: We have been given the diameter instead of the radius, but the formula uses the radius. Since $r=\frac{D}{2}$r=D2, we can determine that the radius is $\frac{2.3}{2}$2.32 or $1.15$1.15 cm.
Do:
$V$V | $=$= | $\frac{4}{3}\pi r^3$43πr3 |
State the formula |
$V$V | $=$= | $\frac{4}{3}\pi\times1.15^3$43π×1.153 |
Fill in the given information |
$V$V | $=$= | $6.370626$6.370626... |
Using a calculator, evaluate using the $\pi$π button |
$V$V | $=$= | $6.37$6.37 |
Round to two decimals |
Find the volume of the sphere shown.
Round your answer to two decimal places.
A sphere has a radius $r$r cm long and a volume of $\frac{343\pi}{3}$343π3 cm3. Find the radius of the sphere.
Round your answer to two decimal places.
Enter each line of working as an equation.
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.