So who cares what the volume of a sphere is. Well...
So apart from having to complete questions at school on volumes of random shapes, volume calculations have a wide variety of applications.
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
Whilst the proof is not typically included as part of the needs of the curriculum, this particular one is a nice introduction to thinking about proofs and abstract proofs. So you don't have to read this if you don't want to but aren't you curious to know where this funny formula came from?
If four points on the surface of a sphere are joined to the centre 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 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 cm^{3}. Find the radius of the sphere.
Round your answer to two decimal places.
Enter each line of working as an equation.
Solve problems involving the surface areas and volumes of prisms, pyramids, cylinders, cones, and spheres, including composite figures