Composite volumes and surface areas
Composite An object with width, height and depth, eg a cube. solids can be created from simple 3D solids.
Example
A salt shaker is made from a cylinder and a Half of a sphere.. Calculate the volume and surface area of the salt shaker.
Total volume of the salt shaker = volume of cylinder + 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 = surface area of cylinder + 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 = \(\pi \times 1.5^2 + 2 \times \pi \times 1.5 \times 5 + \frac{1}{2} \times 4 \times \pi \times 1.5^2 = 68.3~cm^2\)