Most of the formulas in this unit are one idea seen again. A prism's volume is its base area times its height, and a cylinder is a prism with a circle for a base. A cone is a third of the cylinder around it, a pyramid a third of the prism around it, and a sphere is two thirds of the cylinder that just holds it.
The surface is the other half of the story: a cylinder unrolls to a rectangle and two circles, a cone's curved side opens into part of a circle, and a sphere's skin covers exactly four circles. The last lesson puts the solids together and asks a designer's question: for the same amount inside, which shape needs the least material?
Lessons 1, 2, 4 and 6 use a scientific calculator, with a note on the page about the keys. Lessons 3 and 5 need none.
The lessons, in order
- Lesson 1 · Volume of prisms and cylinders Volume of a prism as base area times height, the cylinder as a circular prism, V = πr²h, answers in terms of π and to one decimal place, working backwards to a height, and what doubling does. Stack 1 cm layers to build a cylinder, then double its height, its radius or both and compare the volumes. Scientific calculator
- Lesson 2 · Surface area of a cylinder and its net The net of a cylinder as two circles and a rectangle with width 2πr, closed and open cylinders and pipes, answers in terms of π and to one decimal place, and working back to a height. Unroll a cylinder into its net, then switch the lid and the base off to see how the area changes. Scientific calculator
- Lesson 3 · Pyramids: volume and surface area The volume of a pyramid as one third of base area times height, the slant height of a square pyramid by Pythagoras, the sloping faces and the whole surface, and working back to a height. Fit a pyramid inside its prism and see it is a third, then draw the triangle that gives the slant height. No calculator
- Lesson 4 · Cones: volume and curved surface area The volume of a cone as one third of the cylinder, the slant height by Pythagoras, the curved surface area πrl, the whole surface, and working back to a height. Pour three cone-fuls into a cylinder, then draw the triangle that gives the slant height of a cone. Scientific calculator
- Lesson 5 · Spheres and hemispheres The surface area 4πr² and volume 4/3 πr³ of a sphere, the hemisphere, answers in terms of π, working back to a radius, and the cone, sphere and cylinder in the ratio 1 : 2 : 3. Cover four circles with the peel of a sphere, then set a cone, a sphere and a cylinder side by side. No calculator
- Lesson 6 · Composite solids and choosing a design Composite solids from cylinders, cones and hemispheres, which faces are on the outside, litres and cm³, and comparing designs of equal volume by their surface area. Change the radius of a one-litre tin and watch the metal needed, then build a capsule, an ice-cream cone and a silo. Scientific calculator
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.