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
• 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 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.