Outcomes
A student:

 MA5.31WM
uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

 MA5.32WM
generalises mathematical ideas and techniques to analyse and solve problems efficiently

 MA5.313MG
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
 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 
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 rightangled triangles in threedimensional 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.