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3-dimensional solids - OCRComposite volumes and surface areas

3-dimensional solids have faces, edges and vertices. Volume is the space contained within a 3D solid. Surface area is the sum of the area of each face. 3D solids can be viewed from different points.

Part ofMathsGeometry and measure

Composite volumes and surface areas

Find the refreshed revision resources for GCSE Maths: Surface areas, with step-by-step slideshows, quizzes, practice exam questions, and more!

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

Example

A salt shaker is made from a cylinder and a . Calculate the volume and surface area of the salt shaker.

Diagram showing how to work out the volume of a hemisphere

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 = \(\pi \times 1.5^2 + 2 \times \pi \times 1.5 \times 5 + \frac{1}{2} \times 4 \times \pi \times 1.5^2 = 68.3~cm^2\)