Chapter 12 · Introduction

Chapter 12 — Surface Areas and Volumes

A medicine capsule, a water tank, a tent, an ice-cream cone: real objects are rarely a single cube or sphere. Learn to join two solids and still find the paint they need and the water they hold.

Look at a rocket on the launch pad: a long cylinder topped by a cone. Look at the tank on a petrol tanker: a cylinder closed by a hemisphere at each end. A circus tent is a cylinder wearing a cone as a roof; an ice-cream cone is a cone carrying half a ball. Engineers of ISRO and the people who make tents, pills and water tanks all ask the same two questions: how much material covers the outside? and how much fits inside? The first question is about surface area, the second about volume.

In Class 9 you met the formulas for a single cuboid, cube, cylinder, cone and sphere. This chapter joins two of these solids into one object. The big new idea is gentle but easy to get wrong: when two solids are glued together, the faces that touch disappear from the outside, so surface areas do not simply add — while volumes do simply add (or subtract, when one solid is scooped out of another). Section 12.1 deals with the outside, Section 12.2 with the inside. We use $\pi=\tfrac{22}{7}$ when the radius is a multiple of $7$ (or of $3.5$, $10.5$, $\dots$) and $\pi=3.14$ otherwise, and we always say which.

Single solids and their formulas→Join two solids→Surface: add only the exposed faces→Volume: add or subtract→Cost, capacity, mass
🎯 By the end of this chapter you can…
  • write the curved and total surface area and the volume of a cuboid, cube, cylinder, cone, sphere and hemisphere, and find the slant height of a cone;
  • recognise a combination of two solids and decide which faces stay on the outside;
  • find the total surface area of a cone on a hemisphere, a capsule, a tent, a cube with a hemisphere and other combinations;
  • find the volume of a combination by adding the volumes, or of a solid with a hollow scooped out by subtracting;
  • solve applied problems on cost of painting, canvas for a tent, capacity of a tank and mass of a wooden object;
  • give answers with the value of $\pi$ stated, correct to two decimal places, with units (cm², m², cm³, m³, litres).
Teaser — A Capsule That Grows Longer

A capsule is a cylinder with a hemisphere at each end. Its radius is fixed at $r=3.5$ cm ($\pi=\tfrac{22}{7}$). Slide the length $h$ of the cylindrical part (or press Play). The red curved wall and the two blue hemispheres are the outside; the two circles where they join are hidden. At $h=0$ the capsule is a ball.

8.0
cylinder wall $2\pi rh$ (cm²)—
two ends $4\pi r^2$ (cm²)—
total surface area (cm²)—
volume (cm³)—

Sections in this chapter

12.1

Surface Area of a Combination of Solids

Curved and total surface area of the six basic solids, and how to add only the faces that stay on the outside when two solids are joined or one is scooped out of another.

$\text{CSA}_{\text{cone}}=\pi rl$, $\ l=\sqrt{r^2+h^2}$
Open section →12.2

Volume of a Combination of Solids

Volumes of the basic solids, adding and subtracting them for capsules, tents, cones with ice cream and hollowed blocks, and the 1 : 2 : 3 rule for cone, hemisphere and cylinder.

$V=\tfrac13\pi r^2h+\tfrac23\pi r^3$
Open section →
✅ Before you start
You need the Pythagoras theorem (Class 9) to find a slant height, the area of a circle $\pi r^2$ and the circumference $2\pi r$, and the idea of a sector of a circle from Section 11.1, which explains why the curved surface of a cone has area $\pi rl$. From Class 9 recall that a right circular cylinder or cone has its axis at right angles to the base, and that $1\ \text{m}^3=1000$ litres. Warm up: a hemisphere has diameter $14$ cm. What is its radius? (Answer: $7$ cm.) What is the area of its flat circular face with $\pi=\tfrac{22}{7}$? (Answer: $154$ cm².) We use $\sqrt2\approx1.414$ and $\sqrt3\approx1.732$ if roots arise.
← 11.2 Segments of a Circle12.1 Surface Area of a Combination of Solids →