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NSW Syllabuses

Mathematics K–10 - Stage 5.3 - Measurement and Geometry Area and Surface Area

Outcomes

A student:

  • MA5.3-1WM

    uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

  • MA5.3-2WM

    generalises mathematical ideas and techniques to analyse and solve problems efficiently

  • MA5.3-13MG

    applies formulas to find the surface areas of right pyramids, right cones, spheres and related composite solids

Content

  • Students:
  • Solve problems involving the surface areas of right pyramids, right cones, spheres and related composite solids (ACMMG271)
  • identify the 'perpendicular heights' and 'slant heights' of right pyramids and right cones L
  • apply Pythagoras' theorem to find the slant heights, base lengths and perpendicular heights of right pyramids and right cones
  • devise and use methods to find the surface areas of right pyramids
  • develop and use the formula to find the surface areas of right cones:
    \(\text{Curved surface area of cone}=\pi rl\) where \(r\) is the length of the radius and \(l\) is the slant height CCT
  • use the formula to find the surface areas of spheres:
    \(\text{Surface area of a sphere}=4\pi r^2\) where \(r\) is the length of the radius
  • solve a variety of practical problems involving the surface areas of solids
  • find the surface areas of composite solids, eg a cone with a hemisphere on top (Problem Solving)
  • find the dimensions of a particular solid, given its surface area, by substitution into a formula to generate an equation (Problem Solving)

Background Information

Pythagoras' theorem is applied here to right-angled triangles in three-dimensional space.

The focus in this substrand is on right pyramids and right cones. Dealing with oblique versions of these objects is difficult and is mentioned only as a possible extension.

The area of the curved surface of a hemisphere is \(2 \pi r^2\), which is twice the area of its base. This may be a way of making the formula for the surface area of a sphere look reasonable to students. Deriving the relationship between the surface area and the volume of a sphere by dissection into very small pyramids may be an extension activity for some students. Similarly, some students may investigate, as an extension, the surface area of a sphere by the projection of very small squares onto a circumscribed cylinder.

Language

The difference between the 'perpendicular heights' and the 'slant heights' of pyramids and cones should be made explicit to students.