This chapter develops surface area and volume for important three-dimensional solids. It begins with cuboids and cubes, treating a cuboid as a three-dimensional version of a rectangle. A cuboid has six rectangular faces and is described by its length, width and height. Its total surface area is obtained by accounting for the areas of all six faces, giving TSA = 2(lw + lh + wh).
Nets provide an intuitive way to understand surface area. When the faces of a solid are opened out into a flat arrangement, the total area of the resulting rectangles or other shapes equals the surface area of the solid. This connects a visual model of a solid with an algebraic formula.
The chapter extends these ideas to cylinders, cones, pyramids, spheres and hemispheres. Each solid has a characteristic surface and occupies space, so the relevant formula depends on its dimensions and the geometry of its faces or curved surfaces. The chapter also uses everyday objects and modelling situations to make these measurements meaningful rather than treating formulas as isolated rules.
An important theme is choosing a suitable mathematical model and checking whether the result makes sense. The chapter includes estimation or 'guesstimate' contexts, encouraging students to use reasonable approximations before or alongside exact calculations. Surface area and volume therefore become tools for solving practical problems involving containers, objects and spatial measurements.
For a cylinder of radius r and height h, the curved surface area is 2πrh and the volume is πr²h. If the cylinder is closed at both ends, the two circular ends are added to obtain the total surface area; if it is open at one end, only one circular end is included. The chapter derives the curved surface formula by cutting the curved surface and viewing it as a rectangle, whose width is the circumference of the circular base.
For a right circular cone with radius r, height h and slant height l, the curved surface area is πrl and the total surface area is πr(l + r). Its volume is one-third of the volume of a cylinder having the same base radius and height. For a sphere of radius r, the surface area is 4πr². A hemisphere has curved surface area 2πr², total surface area 3πr², and volume 2/3 πr³. These formulas are used in the chapter together with modelling and estimation activities.