Composite solids

Objective. Find volume and surface area of joined or hollow solids.

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Learn: the key idea

Composite solids are made by joining or removing simple 3D shapes, so their volume and surface area are found by adding or subtracting the component parts' values. Break the solid into recognisable shapes (cuboids, cylinders, cones, hemispheres), calculate each one's volume or surface area separately, then combine appropriately, remembering that joined faces are not part of the external surface area. For example, a cylinder topped with a hemisphere has total volume equal to cylinder volume plus hemisphere volume. For hollow solids, subtract the inner shape's volume from the outer shape's volume.

Worked example

A composite solid is a cylinder of radius 3 cm and height 10 cm with a hemisphere of radius 3 cm on top. Find the total volume, to 1 decimal place.

1 $Cylinder volume = πr²h = π(9)(10) = 90π$
2 $Hemisphere volume = (2/3)πr³ = (2/3)π(27) = 18π$
3 $Total volume = 90π + 18π = 108π$
4 $108π \approx 339.3 cm³$

Answer: 339.3 cm³

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Common mistake. A frequent error is including the internal joining face, like the flat circle between a cylinder and a hemisphere, in the total surface area; only count faces actually exposed to the outside.

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