Chapter 11: Surface Areas and Volumes

The world is full of three-dimensional shapes: boxes, cans, cones, balls. This chapter equips you with formulas for the surface area and volume of the six most common shapes , the cuboid, cube, right circular cylinder, right circular cone, sphere, and hemisphere , and shows you how to apply them to real-life objects.

Each shape gets two formulas: one for surface area (the total area of its outer surface) and one for volume (the amount of space it occupies). Some shapes have additional formulas for the lateral or curved surface area, which is the surface excluding the bases. The chapter is light on theory and heavy on practical computation: identify the shape, plug into the formula, simplify, and compute.

Surface areas and volumes are everywhere: how much paint covers a wall, how much water a tank holds, how much wrapping paper covers a box, how much fuel a tank can store, how much carpet covers a floor. Every such question reduces to choosing the right shape, the right formula, and computing carefully.

A note on $\pi$ : many problems use $\pi \approx \tfrac{22}{7}$ for convenience. Some use the more accurate value $\pi \approx 3.14$ . Always read the question to see which value is requested, and use it consistently throughout the problem.

By the end of the chapter you should be able to compute the surface area and volume of any of the six shapes from given dimensions, identify which shape a real-world object resembles, and combine shapes (e.g. a hemisphere on top of a cylinder) to model composite objects.

What's inside

Cuboid and cube , the simplest 3D shapes, with rectangular faces.
Right circular cylinder , like a can; curved surface + two circular bases.
Right circular cone , like a party hat; curved surface + circular base.
Sphere , perfect 3D symmetry, like a ball.
Hemisphere , half a sphere; curved surface + flat circular base.

Key results / Formula card

Shape	Surface area	Volume
Cube (side $a$ )	$6a^2$	$a^3$
Cuboid (length $\ell$ , breadth $b$ , height $h$ )	$2(\ell b + bh + h\ell)$	$\ell b h$
Cylinder (radius $r$ , height $h$ )	Curved: $2\pi r h$ ; Total: $2\pi r(h+r)$	$\pi r^2 h$
Cone (radius $r$ , height $h$ , slant $\ell$ )	Curved: $\pi r \ell$ ; Total: $\pi r (\ell+r)$	$\tfrac{1}{3}\pi r^2 h$
Sphere (radius $r$ )	$4\pi r^2$	$\tfrac{4}{3}\pi r^3$
Hemisphere (radius $r$ )	Curved: $2\pi r^2$ ; Total: $3\pi r^2$	$\tfrac{2}{3}\pi r^3$
Cone slant height	$\ell = \sqrt{r^2 + h^2}$

Memorise this card. Almost every problem in the chapter is a direct application of one of these formulas.

Chapter 11: Surface Areas and Volumes

What's inside

Key results / Formula card

Sub-topics

Practice quiz