0580

Volume and Surface Area

Lengths, Areas and Volumes

Building it from the faces

  • Surface area is the total of every face and curved surface on the outside of the solid
  • For a solid with flat faces, work out each face separately and add them, using the net if it helps
    • A cuboid 7 cm by 4 cm by 3 cm has faces in three matching pairs: 2(7 × 4) + 2(7 × 3) + 2(4 × 3) = 56 + 42 + 24 = 122 cm²
    • A square-based pyramid has one square and four identical triangles
The net of a cube laid flat as six squares in a cross, so the surface area is just the total area of those six faces
Source: Equations & problem-solving by Save My Exams
  • For a cylinder, add the curved surface to the two circular ends: 2πrh + 2πr²
  • For a cone, add the curved surface to the circular base: πrl + πr²
A cone labelled with its radius r, perpendicular height h and slant height l along the sloping edge, beside its net: a sector of slant radius l joined to a circle of radius r, so the curved surface uses l and not h
Source: Surface Area by Save My Exams
  • Only the curved parts are printed, so these totals must be assembled
  • Check the units: surface area is always in squared units even though the solid is three-dimensional
Exam tip

Work along the outside of the solid and list each face you would actually touch, then add only those. Only the curved surfaces are printed in the List of formulas, so the flat ends of a cylinder or a cone have to be added by hand, and a face where two solids join is not a surface at all.

Worked example

Total surface area of a cylinder

A solid cylinder has radius 5 cm and height 12 cm. Find its total surface area in terms of π.

An upright solid cylinder drawn with a full ellipse for the visible top rim and a dashed far half on the bottom rim. A dashed radius on the top face is labelled 5 cm and the vertical side is labelled 12 cm. The figure is marked NOT TO SCALE.

Solution:

  • A solid cylinder has three surfaces: two circular ends and the curved side
  • The curved surface area is the one that is printed on the paper: 2πrh
  • Curved surface = 2 × π × 5 × 12 = 120π
  • Each circular end has area πr² = π × 5² = 25π
  • There are two of them, giving 2 × 25π = 50π
  • Total surface area = 120π + 50π = 170π cm²
  • The total for a cylinder is not printed, only the curved part, so the two ends have to be added by hand
  • If the cylinder were open at one end, only one circle would be added