Surface Areas and Volumes

Solids made by joining cylinders, cones, hemispheres and cubes: add the volumes, and add only the surfaces that stay visible.

Key ideas

The volume of a combined solid is the sum of the volumes of its parts.
The surface area of a combined solid is the sum of the surfaces you can still see; the faces where parts are joined are hidden.
A hemisphere on a cylinder hides the top of the cylinder and the flat face of the hemisphere.
When a part is scooped out, subtract its volume; for the area, take away the face that was cut and add the inside surface.
Slant height of a cone: l = √(r² + h²).
Convert units at the end: 1000 cm³ = 1 litre, 1 m³ = 1000 litres.

Remember

Cylinder	curved area 2πrh; volume πr²h
Cone	curved area πrl; volume (1/3)πr²h
Hemisphere	curved area 2πr²; volume (2/3)πr³
Sphere	area 4πr²; volume (4/3)πr³

Watch out

Adding the total surface areas of the parts, so the joined faces are counted.
Using the height of a cone where its slant height is needed (or the other way round).
Mixing cm and m in one calculation.

Worked example

A hemispherical hollow of radius 7 cm is scooped out of the top of a solid wooden block 33 cm × 32 cm × 15 cm. Find the volume of wood left (π = 22/7).

Working: 15,840 − (2/3) × 22/7 × 7³ = 15,840 − 718 2/3

Answer: 15,121 1/3 cm³