CONES, PYRAMIDS AND SPHERES - Measurement and Geometry Module 12 - A guide for teachers-Years 9-10

Sep 18, 2026

Uploaded by Mabel

Teacher-focused module from the Improving Mathematics Education in Schools (TIMES) Project covering measurement and geometry for Years 9–10. The guide reviews assumed knowledge and builds motivation using prism volume and surface area formulas, then extends methods to non-uniform solids. It develops the surface area of a cylinder via unfolding into a rectangle, and continues toward volume and surface area formulas for a pyramid, a cone and a sphere, illustrated with worked examples and reference diagrams.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages24

Reading Time1 h 0 min

AccessFree to download

ASSUMED KNOWLEDGE

MOTIVATION

SURFACE AREA OF A CYLINDER

Example: Surface area of each solid

The Improving Mathematics Education in Schools (TIMES) Project CONES, PYRAMIDS AND SPHERES A guide for teachers-Years 9–10 MEASUREMENT AND GEOMETRY Module 12 June 2011 Cones, Pyramids and Spheres (Measurement and Geometry : Module 12) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated) . This work is licensed under the Creative Commons Attribution‑NonCommercial‑NoDerivs 3.0 Unported License. 2011. [http://creativecommons.org/licenses/by](http://creativecommons.org/licenses/by)‑nc‑nd/3 .0/ The Improving Mathematics Education in Schools (TIMES) Project CONES, PYRAMIDS AND SPHERESA guide for teachers-Years 9–10 MEASUREMENT AND GEOMETRY Module 12 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 910 {4} A guide for teachers CONES, PYRAMIDS AND SPHERES ASSUMED KNOWLEDGE • Familiarity with calculating the areas of the standard plane figures including circles. • Familiarity with calculating the volume of a prism and a cylinder. • Familiarity with calculating the surface area of a prism. • Facility with visualizing and sketching simple three‑dimensional shapes. • Facility with using Pythagoras’ theorem. • Facility with rounding numbers to a given number of decimal places or significant figures. • Facility with manipulation of formulas and equations MOTIVATION In the earlier module, Area Volume and Surface Area we developed formulas and principles for finding the volume and surface areas for prisms. The volume of a prism, whose base is a polygon of area A and whose height is h, is given by Volume of a prism = Ah. This formula is also valid for cylinders. Hence, if the radius of the base circle of the cylinder is r and its height is h, then: Volume of a cylinder = πr2h Also in that module, we defined the surface area of a prism to be the sum of the areas of all its faces. For a rectangular prism, this is the sum of the areas of the six rectangular faces. For other prisms, the base and top have the same area and all the other faces are rectangles. In this module, we will examine how to find the surface area of a cylinder and develop the formulae for the volume and surface area of a pyramid, a cone and a sphere. These solids differ from prisms in that they do not have uniform cross sections. This will complete the discussion for all the standard solids. The Improving Mathematics Education in Schools (TIMES) Project {5} Pyramids have been of interest from antiquity, most notably because the ancient Egyptians constructed funereal monuments in the shape of square based pyramids several thousand years ago. Conical drinking cups and storage vessels have also been found in several early civilisations, confirming the fact that the cone is also a shape of great antiquity, interest and application. The word sphere is simply an English form of the Greek sphaira meaning a ball. Conical and pyramidal shapes are often used, generally in a truncated form, to store grain and other commodities. Similarly a silo in the form of a cylinder, sometimes with a cone on the bottom, is often used as a place of storage. It is important to be able to calculate the volume and surface area of these solids. The ancient Greeks discovered the various so‑called quadratic curves, the parabola,

FAQs

What prior topics do students need for this module?

Students are expected to know areas of standard plane figures (including circles), volumes and surface areas of prisms, how to visualize and sketch 3D shapes, apply Pythagoras’ theorem, round numbers accurately, and manipulate formulas and equations.

How is the surface area of a cylinder derived in the module?

By rolling the cylinder one revolution, the curved surface traces out a rectangle of width h and length 2πr. The curved area is 2πrh, and adding the two circular ends gives 2πrh + 2πr².

What solids are introduced beyond prisms in this module?

The module explains how cylinders, pyramids, cones, and spheres differ from prisms because they do not have uniform cross sections, then develops corresponding volume and surface area formulas.

Sep 18, 2026

Uploaded by Mabel