Quick Revision

Cuboid and Cube

Cuboid: volume $lbh$ , total surface $2(lb + bh + hl)$ , four walls $2h(l + b)$ , diagonal $\sqrt{l^2 + b^2 + h^2}$ .
$(l + b + h)^2 = d^2 + \text{total surface}$ ; the 12 edges add up to $4(l + b + h)$ .
Faces $x$ , $y$ , $z$ at a corner: volume $\sqrt{xyz}$ .
Cube: $a^3$ , $6a^2$ , four walls $4a^2$ , diagonal $a\sqrt{3}$ , face diagonal $a\sqrt{2}$ .
Cut into cubes of edge $b$ : $\left(\frac{a}{b}\right)^3$ pieces, surface × $\frac{a}{b}$ . $n$ cubes in a row: $(4n + 2)a^2$ .
A cube's surface $2d^2$ needs the full diagonal $a\sqrt{3}$ .

Four walls = perimeter of the floor × height, less doors and windows.
Paper or carpet of width $w$ : length = area ÷ $w$ .
Squares of side $x$ cut from an $L \times B$ sheet: box $(L - 2x)(B - 2x)x$ .
Wood $t$ thick: a closed box loses $2t$ from each outside measure; an open box loses only $t$ from its height.
Bricks = wall volume (less mortar) ÷ one brick.
1 m³ = 1000 litres and 1 litre = 1000 cm³. Rain = area × depth in metres.

Volume = base area × height.
Lateral surface = perimeter of the base × height; total = lateral + 2 × base.
Bases you will meet (see Chapter 13): equilateral triangle $\frac{\sqrt{3}}{4}a^2$ , right triangle $\frac{1}{2}ab$ , rhombus $\frac{1}{2}d_1d_2$ , trapezium, regular hexagon $\frac{3\sqrt{3}}{2}a^2$ .
Rhombus side from the half-diagonals: $\frac{1}{2}\sqrt{d_1^2 + d_2^2}$ .
An $n$ -sided base: $n + 2$ faces, $3n$ edges, $2n$ vertices. Euler: $F + V - E = 2$ .
A prism has two bases, and both belong in the total surface.

Volume $= \frac{1}{3} \times \text{base} \times \text{height}$ .
Lateral surface $= \frac{1}{2} \times \text{perimeter} \times \text{slant height}$ ; total = lateral + one base.
Square base of side $a$ : slant height $l$ with $l^2 = h^2 + \frac{a^2}{4}$ ; edge to a corner $e$ with $e^2 = h^2 + \frac{a^2}{2}$ .
An $n$ -sided base: $n + 1$ faces, $2n$ edges, $n + 1$ vertices.
[Tier 2] Equilateral base: the centre is $\frac{a}{2\sqrt{3}}$ from each side and $\frac{a}{\sqrt{3}}$ from each corner.
[Tier 2] Regular tetrahedron: surface $\sqrt{3}a^2$ , height $a\sqrt{\frac{2}{3}}$ , volume $\frac{a^3}{6\sqrt{2}}$ .

Volume $\pi r^2h$ , curved surface $2\pi rh$ , total $2\pi r(r + h)$ .
Volume ÷ curved surface $= \frac{r}{2}$ , a quick way to the radius.
Two cylinders: volumes go as $r^2h$ , curved surfaces as $rh$ .
A sheet rolled up: the side that goes round is $2\pi r$ , the other side is the height.
Roller: area pressed = curved surface × number of turns.
Tank open at the top: $2\pi rh + \pi r^2$ .
Before squaring anything, check whether the question gave the radius or the diameter.

Metal $= \pi(R^2 - r^2)h = \pi(R + r)(R - r)h$ ; the thickness is $R - r$ .
Outer surface $2\pi Rh$ , inner $2\pi rh$ , each rim $\pi(R^2 - r^2)$ ; all of them together $2\pi(R + r)(h + R - r)$ .
Flow through a pipe each second $= \pi r^2 \times$ speed; km/h × $\frac{5}{18}$ gives m per second.
[Tier 2] Well and embankment: $\pi r^2H = \pi(R^2 - r^2)h$ , with $R = r$ + width.
$\pi(R - r)^2h$ is a thin rod, not the pipe.

Slant height $l = \sqrt{r^2 + h^2}$ . Triplets: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29.
Curved surface $\pi rl$ , total $\pi r(l + r)$ , volume $\frac{1}{3}\pi r^2h$ .
On the same base and height as a cylinder, a cone holds one third as much.
Sector into cone: the sector's radius is $l$ and its arc is $2\pi r$ , so $\frac{r}{l} = \frac{\theta}{360^\circ}$ .
Tent or heap: canvas = curved surface only; length of canvas = area ÷ width.
Volume takes the height, surface the slant: mark $h$ and $l$ on your sketch.

Cut parallel to the base: the top cone has lengths $k$ , surfaces $k^2$ , volume $k^3$ .
Halfway cut: top : frustum = 1 : 7. Three equal heights: 1 : 7 : 19.

$Frustum with radii R and r height h slant l$

Volume $\frac{\pi h}{3}(R^2 + Rr + r^2)$ ; slant $\sqrt{h^2 + (R - r)^2}$ .
Curved surface $\pi(R + r)l$ ; the total adds $\pi R^2 + \pi r^2$ , an open bucket only $\pi r^2$ .
The slant uses $R - r$ , never $R + r$ .

Sphere: volume $\frac{4}{3}\pi r^3$ , surface $4\pi r^2$ .
Hemisphere: volume $\frac{2}{3}\pi r^3$ , curved surface $2\pi r^2$ , solid total $3\pi r^2$ .
Two spheres: surfaces as $r^2$ , volumes as $r^3$ .
Shell: $\frac{4}{3}\pi(R^3 - r^3)$ . Bowl: holds $\frac{2}{3}\pi r^3$ ; inside surface $2\pi r^2$ .
Sphere in a cube: $r = \frac{a}{2}$ . Cube in a sphere: $a\sqrt{3} = 2R$ .
In a cylinder of height $2r$ : volumes 2 : 3, and the sphere's surface equals the cylinder's curved surface.
Cone, hemisphere and cylinder of radius $r$ and height $r$ : 1 : 2 : 3.

$Toy with the joint circle left out of the surface$

Toy: $\pi rl + 2\pi r^2$ . Tent: $2\pi rh + \pi rl$ .
Capsule: $2\pi r(h + 2r)$ , which is $\pi \times$ width × full length.
Hemisphere scooped from a cube: $6a^2 + \pi r^2$ (the bowl adds $2\pi r^2$ , the hole removes $\pi r^2$ ).
Volumes add, or subtract what is cut out.
A tent has no floor; a solid toy does.

Volume before = volume after. Surface area is not kept.
Number of pieces = big volume ÷ volume of one piece. Cancel $\pi$ and the thirds first.
Spheres into spheres: $\left(\frac{R}{r}\right)^3$ . Several spheres into one: $R^3 = r_1^3 + r_2^3 + r_3^3$ .
Sphere into a wire: length $\frac{4R^3}{3r^2}$ .
Rise in water = volume put in ÷ base area of the vessel.
A hollow solid gives only its metal, $R^3 - r^3$ .
If 10% is wasted, recast the 90% that is left.

Every length × $k$ : surfaces × $k^2$ , volume × $k^3$ .
Every length up 10%: surfaces up 21%, volume up 33.1%. Up 20%: 44% and 72.8%. Down 20%: down 36% and 48.8%.
Different changes: multiply. Radius up 10%, height down 10%: $1.1^2 \times 0.9 = 1.089$ , so a cylinder's volume goes up 8.9%.
Same volume with the radius × $k$ : the height is divided by $k^2$ .
Backwards, take the cube root: volume × 1.331 means lengths × 1.1, which is 10%, not $\frac{33.1}{3}$ .

Put every length in one unit. For litres: m³ × 1000, or cm³ ÷ 1000.
Ask three questions: radius or diameter, height or slant, curved or total surface.
Pick $\pi$ from the options: multiples of 11 or 22 mean $\frac{22}{7}$ .
Size checks: a cone is a third of its cylinder, and a sphere fills about half of the cube around it.
For a combined solid, sketch it and tick each visible surface once.
Then work through the PYQ Practice Sets of this chapter, keeping each solids question under a minute.