Composite volumes and surface areas

Composite 3D shapes can be created from simple 3D shapes.

Example

A salt shaker is made from a cylinder and a hemisphere. Calculate the volume and surface area of the salt shaker. (Ignore the holes!)

Diagram showing how to work out the volume of a hemisphere

Total volume of the salt shaker = \text{volume of cylinder} + \text{volume of hemisphere}

Volume of cylinder =  \pi r^2 h = \pi \times 1.5^2 \times 5

A hemisphere is half a sphere.

Volume of a hemisphere = \frac{1}{2} \times \frac{4}{3} \times \pi r^3 = \frac{1}{2} \times \frac{4}{3} \times \pi \times 1.5^3

Total volume of the salt shaker = \pi \times 1.5^2 \times 5 + \frac{1}{2} \times \frac{4}{3} \times \pi \times 1.5^3 = 42.4~\text{cm}^3

Total surface area of the salt shaker = \text{surface area of cylinder} + \text{surface area of hemisphere}

Surface area of cylinder (note only one circular end) = \pi r^2 + 2\pi rh = \pi \times 1.5^2 + 2 \times \pi \times 1.5 \times 5.

A hemisphere is half a sphere.

Curved surface area of hemisphere = \frac{1}{2} \times 4\pi r^2 = \frac{1}{2} \times 4 \times \pi \times 1.5^2.

Total surface area of the salt shaker = \pi \times 1.5^2 + 2 \times \pi \times 1.5 \times 5 + \frac{1}{2} \times 4 \times \pi \times 1.5^2 = 68.3~cm^2