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Area and Volume
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Last sentence first
- Start:
- Start at the last sentence to see if you need area, volume, a length or surface area.
- Stop:
- Stop reading the story once you have every number with its name.
- Skip:
- Skip the decoration about tanks, cans and boxes.
- Why:
- The setting never changes the formula. Only the shape and the asked quantity matter.
Used in questions 1, 2, 3, 7
2. Radius before square
- Start:
- Find r (halve any diameter) and write it down alone.
- Stop:
- Stop after you have written r² as a number.
- Skip:
- Do not multiply by π or h until r² is done.
- Why:
- Squaring the whole product is the most common wrong turn, as in the 900π trap.
Used in questions 2, 3, 6
3. Height is straight up
- Start:
- Look for the word 'height' or a right-angle mark.
- Stop:
- Stop when you have a height, not a slanted side.
- Skip:
- Ignore the slant length for volume, unless you need it for surface.
- Why:
- Volume and area formulas use perpendicular height, so the slant distracts.
Used in questions 4, 6
4. Pouring means equal volume
- Start:
- Read which container is first and find its volume.
- Stop:
- Stop when you set that volume equal to the second container's formula.
- Skip:
- Skip the unit labels until the end.
- Why:
- Water does not disappear, so volume stays the same and only the shape changes.
Used in question 7
5. Ratio shortcut for scaling
- Start:
- Compare the new formula to the old one letter by letter.
- Stop:
- Stop when only a plain number multiplier is left.
- Skip:
- You do not need to know r or h themselves.
- Why:
- The unknown letters cancel, and you only multiply the old volume by the leftover number (here 3).
Used in questions 8, 3
6. Check with a calculator when the result has π
- Start:
- In Desmos or the Bluebook calculator, type the formula directly.
- Stop:
- Stop when the decimal matches an option's decimal value.
- Skip:
- Do not simplify by hand if options are close.
- Why:
- Typing 3π ≈ 9.42 and comparing to 4π ≈ 12.57 stops small mistakes.
Used in questions 2, 3, 5, 7
7. Powers of the scale factor
- Start:
- Find the scale factor k between the two solids from any pair of matching lengths.
- Stop:
- Stop when you decide: lengths use k, areas use k², volumes use k³.
- Skip:
- Skip computing the new edges one by one.
- Why:
- A 5/3 scale factor gives 125/27 for volume, so 27 becomes 125 in one step.
Used in questions 14, 8
8. Whole minus hole
- Start:
- Find the volume of the full shape, then the volume of the removed part.
- Stop:
- Stop after you subtract; leave π in the answer if the options have π.
- Skip:
- Do not mix side area (2πrh) with volume (πr²h).
- Why:
- A hole is a cylinder with its own radius and height, so two simple formulas are safer than one long one.
Used in questions 11, 13
9. Raised water equals pushed-in volume
- Start:
- Find the volume of the object that goes into the water, then the base area of the tank.
- Stop:
- Stop after you divide volume by base area; this is the rise in depth.
- Skip:
- Skip the starting depth of the water; it does not change the rise.
- Why:
- Water that is moved up must fill the same volume as the object, spread over the tank's base. The same idea fits filling a tank at a steady rate (volume ÷ rate = time).
Used in questions 15, 12