MathematicsExam code: 0580

Volume and Surface Area

Lengths, Areas and Volumes

PractiseFlashcards

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

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

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