A sphere is a solid 3D round object whose surface is the collection of all points that are equidistant from a central point (centre of the sphere). Half of a sphere is called a hemisphere.
Surface area of a sphere
Unlike the solids we have seen so far, we cannot unwrap a sphere to get a 2D net to calculate its area. So the surface area of a sphere is approached in a different way.
Archimedes discovered that for a cylinder that circumscribes a sphere as shown below, the area of the curved face of the cylinder is equal to the surface area of the sphere. This is not easy to prove!
We saw above that the curved surface of a cylinder flattens out into a rectangle. Since the height of this cylinder is twice the radius of the inscribed sphere, which is also the radius of the circular ends, the area of the resulting rectangle is $2\pi r\times2r=4\pi r^2$2πr×2r=4πr2 . We therefore have our formula for the surface area of a sphere.
$SA=4\pi r^2$SA=4πr2
To find the surface area of a hemisphere, we consider that the curved face has half the surface area of a sphere, and the flat face is a circle of radius $r$r. So the surface area of a hemisphere will be:
| $SA$SA | $=$= | $\frac{1}{2}\times4\pi r^2+\pi r^2$12×4πr2+πr2 |
| $=$= | $2\pi r^2+\pi r^2$2πr2+πr2 | |
| $=$= | $3\pi r^2$3πr2 |
Practice Questions
Question 1
The planet Mars has a radius of $3390$3390 km. What is the surface area of Mars?
(Give your answer correct to the nearest whole km2.)
Question 2
Consider the following hemisphere with a radius of $8$8. Find the total surface area.
Round your answer to three decimal places.
Volume of a sphere
One way to consider the volume of a sphere is to approximate it as the composition of lots of pyramids with apexes that meet at the centre of the sphere.
The volume of a pyramid is $\frac{1}{3}Ah$13Ah. Since each pyramid has its apex at the centre of the sphere, they will all have a height of $r$r, the radius of the sphere. And to get the best approximation, we want the areas of the bases of the pyramids to sum to the surface area of the sphere, which is $4\pi r^2$4πr2.
As such, the volume of a sphere with radius $r$r can be calculated using the following formula:
| Volume of a sphere | $=$= | $\frac{1}{3}\times4\pi r^2\times r$13×4πr2×r |
| $=$= | $\frac{4}{3}\pi r^3$43πr3 |
If we are asked to find the volume of a hemisphere, we would find the volume of a sphere with the same radius, then halve the result.
Practice questions
Question 3
Find the volume of the sphere shown.
Round your answer to two decimal places.
