Spheres - Higher tier only
A sphere is a perfectly round solid figure. All points on the surface of the shape are the same distance away from the centre 鈥 we call this distance the radius.
The formula for the volume and surface area of a sphere will be given to you in the exam, so you will not need to memorise these.
\(\text{The volume of a sphere} ~=~ \frac {4}{3} \times \pi \times \text{r}^{3}\)
\(\text{The surface area of a sphere} ~=~ \text{4} \times \pi \times \text{r}^{2}\)
The diameter is the distance from one point on the surface to another, through the centre. If you are given the diameter you must divide by 2 to find the radius before you can calculate the volume or surface area.
Example
A spherical fish tank of diameter 40 cm is half full of water.
1. Calculate the volume of the water
2. The water is transferred into a new spherical tank so that the water fills the tank completely. Calculate the surface area of the new tank
Solution
1. Calculate the volume of the tank:
Diameter = 40 cm so the radius is 40 梅 2 = 20 cm
Substitute this into the formula for the volume of a sphere:
Volume = \(\frac{4}{3} \times \pi \times \text{r}^{3}\) = \(\frac{4}{3} \times \pi \times ~\) 203 = 33,510.32164 cm3
The tank is half full so dividing by 2 calculates the volume of the water:
33,510.32164 梅 2 = 16,755.16082 cm3
2. Calculate the radius of the new tank, and then find the surface area:
Volume = \(\frac{4}{3} \times \pi \times \text{r}^{3}\) = 16,755.16082
To find \(\text {r}\) we will need to rearrange the formula:
Divide both sides by \(\frac{4}{3} ~ \pi\):
16,755.16082 \(\div \frac{4}{3} ~ \pi ~=~ \text{r}^{3}\)
4,000 = \(\text{r}^{3}\)
Then take the cubed root of both sides:
\(\sqrt[3]{4,000} ~=~ \text{r}\)
\(\text{r} ~\) = 15.87401052 cm
We can now calculate the surface area:
\(\text{4} ~ \pi ~ \text{r}^{2} = {4} \times \pi \times\) 15.874010522 = 3166.526972 cm2
Surface area = 3,166.53 cm2 (to two decimal places)
Question
A rubber band ball has a radius of 6 cm.
I add some more rubber bands and the volume increases by 100 cm3. How much has the radius increased by?
Give your answer to two decimal places.
Old volume = \(\frac{4}{3}~{蟺}~{r^3}~=~\frac{4}{3}~\times~{蟺}~\times~\) 63 = 904.7786842 cm3
New volume = 904.7786842 + 100 = 1,004.7786842 cm3
1,004.7786842 = \(\frac{4}{3}~{蟺}~{r^3}\)
1,004.7786842 \(\div~\frac{4}{3}~{蟺}~=~{r^3}\)
\(\sqrt[3]{239.8732415}~=~{r}\)
\({r}\) = 6.213370741 cm = 6.21 cm (to 2 decimal places)
The radius has increased by 0.21 cm
Question
A lipstick container has a diameter of 12 mm and a height of 52 mm. It consists of a hemisphere on top of a cylindrical tube. Calculate the surface area of the lipstick container.
Cylinder:
Radius = 12 梅 2 = 6 mm
\(\text{Area of base} = 蟺 \times r^2\)
Area of base = \({蟺}\) x 62 = 113.0973355 mm2
\(\text{Area of curved surface} = \text{circumference} \times \text{height}\)
\(\text{Circumference} = 2~蟺~r\)
Circumference = 2 x \(蟺\) x 6 = 37.69911184 mm2
Height = 52 鈥 6 = 46 mm
Area of curved surface = 37.69911184 x 46 = 1734.159145 mm2
Hemisphere:
\(\text{Surface area of sphere} = 4 \times 蟺 \times r^2\)
This formula is given to you in the exam.
Surface area of sphere = 4 x \(蟺\) x 62 = 452.3893421 mm2
Surface area of hemisphere = 452.389 梅 2 = 226.1946711 mm2
Total surface area = base + curved surface + hemisphere = 113.0973355 + 1734.159145 + 226.1946711 = 2073.4511516 mm2
Total surface area = 2,073 mm2 (nearest mm2)
Hemispheres
A hemisphere is exactly half of a sphere. When calculating the volume you would need to halve the volume of a sphere.
When finding the surface area, in addition to halving the surface area of a sphere you will also need to calculate the area of the circle at the base.
Question
Calculate the surface area of the hemisphere that has a diameter of 8 m.
Radius = 4 m
Surface area of dome = \(\frac{1}{2} \times {4} \times {蟺} \times {4^2}~=~{100.5309649}~m^2\)
Area of circular base = \({蟺} \times {r^2}~=~{蟺} \times {4^2}~=~{50.26548246}~m^2\)
Total surface area = 100.5309649 + 50.26548246 = 150.7964474 m2
Total surface area = 150.8 m2 (to one decimal place)
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