The full set of rules, formulas and methods — stripped of worked examples — so you can sit and learn the material.
1. Shape and information — what is it, what are you told?
2. Formula — write the right one down first.
3. Put the figures in and work it out.
4. Units are vital.
Straight down from the tip to the base — not the slanted side.
This is the H in the triangle area formula.
Area A = ℓ × b
Perimeter P = 2(ℓ + b)
Base & perpendicular height: A = 12BH In Tables
Two sides & the angle between: A = 12ab sin C In Tables
One corner at the origin: A = 12|x1y2 − x2y1|
A = a h (base × perpendicular height) or A = a b sin C In Tables
Same idea as the triangle — but no half, because a parallelogram is two triangles put together.
Sine gives two angles between 0° and 180°: the acute one and its partner 180° − C.
A parallelogram can be drawn either way, so both are correct.
Area A = πr2 In Tables
Circumference ℓ = 2πr In Tables
The distance round a circle is the circumference — that's the perimeter, the length, all the same thing.
π = 3.14 or π = 227 — use whichever the question asks for.
"In terms of π" means don't multiply π out — leave the symbol in the answer.
Area A = πr2(θ360) In Tables
Arc length = 2πr (θ360) In Tables
θ is "theta", a Greek letter. 360° is the full circle.
Area A = 12r2θ In Tables
Arc length = rθ In Tables
The arc is the curved edge. The perimeter of a sector is the arc plus the two straight radii: 2r + ℓ.
In radians the arc is ℓ = rθ.
A reflex sector uses the same formulas — the angle is just bigger than 180°. Nothing changes.
Shapes joined together → add the areas.
One cut out of another → Outside − Inside.
Method: split it into rectangles, triangles or a circle, find each piece, then add them or subtract the cut–out.
Step 1. Write the area as a formula.
Step 2. Use the given information to get it down to one variable.
Step 3. Differentiate and let dAdx = 0.
Step 4. Solve for the variable, then find the area.
The total area of all the faces of a solid.
Volume V = ℓ × b × h In Tables
Surface area A = 2(ℓb + ℓh + bh)
Volume V = πr2h In Tables
Curved area A = 2πrh In Tables
Each end circle A = πr2
Open top means there is no lid — so leave that face out.
Open–top cylinder = curved part + one circle (the base).
Solid cylinder = curved part + two circles.
Slant height ℓ2 = h2 + r2
Volume V = 13πr2h In Tables
Curved area A = πrℓ In Tables
Volume V = 43πr3 In Tables
Surface area A = 4πr2 In Tables
Volume V = 23πr3
Curved area A = 2πr2
Solid total A = 2πr2 + πr2 = 3πr2
When it is solid, add the flat circle on top (πr2) to the curved part (2πr2).
Write the volume (or area) formula, put the known value in, and keep the unknown on the left.
Then solve for the missing length.
A solid opened out flat.
Six rectangles.
Two circles (the ends) and one rectangle (the curved part).
The rectangle's height is the cylinder's height; its width is the circumference, 2πr.
One circle (the base) and one sector (the curved part).
The sector's straight edge is the slant height ℓ; its arc is the base circumference, 2πr.
Volume: just add the two volumes.
Surface area: add only the surfaces you can actually see — drop any face that's hidden where the two solids meet.
Step 1. When a solid is melted down and recast, the volume stays the same.
Step 2. Set Vold = Vnew and solve for the missing length.
Step 3. "How many can be made?" — divide, then round down to a whole shape.
Empty space = container − solid.
% = spacecontainer × 100
Draw a front view (2–D). It tells you the container's dimensions at a glance.
A ball that just fits a box → box side = diameter.
A ball that just fits a cylinder → height = diameter, and the cylinder radius = the ball's radius.
Drop an object in — the water level rises.
The risen band of water is a cylinder: same radius as the container, and the same volume as the object you dropped in.
Volume per second = πr2 × speed (a cylinder one second long).
Time = total volumevolume per second
Length is in m, area is in m2, volume is in m3.
1 litre = 1000 cm3
1 m = 100 cm
1 m2 = 10 000 cm2
1 m3 = 1 000 000 cm3
Both units must be the same before you start.
If lengths differ, convert one — e.g. 80 cm = 0.8 m.
If you're given litres, change them to cm3 first so the units match.