4.6 J) Spheres – Part 2
We work out the volume of part of sphere by multiplying the proportion of the sphere that we have by the volume of the whole sphere.
The method for working out the surface area for part of a sphere is slightly different. We work out the curved surface area in a similar way to the volume; we multiply the proportion of the sphere that we have by the surface area of the whole sphere. We then need to work the surface area for the parts of the sphere that are on show. The parts of the sphere that are on show depends on how the part of the sphere has come about; this will all make sense after we have a look at an example.
The shape below is one eighth of a sphere. What is the volume and surface area of this shape? Give your answers to one decimal place.
The volume of the eighth of a sphere is 381.7 cm3.
Area
The area of the eighth of one sphere is worked out by adding the areas of all of the faces together. There are 4 faces on the shape below; the curve surface and the 3 quarter circles.
We work out the area of the curved surface in a similar way to the volume; we find the surface area of the whole sphere and then multiply by the proportion of the whole sphere that we have. We find the surface area of the whole sphere by subbing in r as 9 into the formula below:
We now multiply by one eighth.
This is not our final answer, so we should keep it in exact form for the time being; it is best not to round.
The other 3 faces are each a quarter of a whole circle. This means that the total area of these 3 faces is three quarters of a full circle. Therefore, we can work out the area of these 3 faces by multiplying three quarter by the area of a full circle.
The final step is to add the area of the curved surface to the area of the 3 quarter circles.
We now round to one decimal place.
The total surface area of the shape is 318.1 cm2.