The most common Geometry question type on the SAT (5.3% of the bank). Covers using the provided reference sheet rather than memory, keeping units straight between linear, square and cubic measures, the scaling rule that areas grow by the square and volumes by the cube, composite shapes built by adding and subtracting whole shapes, sectors and arcs as fractions of a circle, and working backwards from a volume to a missing dimension — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
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Title
SAT Math · Type 8 of 19
5.3% of the question bank — 88 of 1675 questions
Objectives
Shapes and solids. Every formula you could need is one tap away inside the testing app, so this type is not a memory test at all. What it actually tests is whether you keep your units straight, whether you know how area and volume respond when lengths change, and whether you can take a formula apart to find a dimension rather than a total.
One sentence is worth more than any formula in this deck: doubling every length multiplies area by four and volume by eight.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 88 tagged questions of this type in the site's bank
Section
Section 1
Concept
Shapes and solids. Every formula you could need is one tap away inside the testing app, so this type is not a memory test at all. What it actually tests is whether you keep your units straight, whether you know how area and volume respond when lengths change, and whether you can take a formula apart to find a dimension rather than a total.
You will see it phrased in these ways:
The third phrasing carries far more weight than its frequency suggests, because almost nobody has been taught the rule explicitly and the intuitive answer is always wrong.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): Area and volume: the tell, the move, and the trap
Notice that the green panel starts by telling you to look something up. On this type, memorising is the wasted effort and reading is the skill.
Prediction
Knowing what the reference sheet carries is worth more than knowing the formulas.
Predict first
Three of these are provided in the app. Which must you know yourself?
Correct: The formula for percent change
Why: The reference sheet is a geometry sheet. It carries circle and triangle formulas, the volumes of the standard solids, the special right triangles and the Pythagorean theorem. It carries nothing algebraic — no percent change, no slope, no quadratic formula, no exponent rules. So the effort of memorising geometry formulas is wasted, and the algebra you do need is exactly what is not there.
Concept
Four shapes of question sit under this label, and they get progressively less familiar.
| variant | what to do | how often |
|---|---|---|
| Direct substitution | read the formula, put the numbers in | most common |
| Composite figure | add or subtract whole shapes | common |
| Working backwards | rearrange the formula for a dimension | common |
| Scaling | apply the k, k squared, k cubed rule | less common, rarely answered correctly |
The first variant is nearly free once you open the reference sheet. The last is where a prepared student gains a mark over an unprepared one.
Definition probe
Units are the fastest check on whether you have used the right formula.
Sort into buckets
What units does each quantity have?
Discrimination
Naming the formula is most of the work.
Sort into buckets
What are you being asked for?
Warm-up
Try it on instinct, then see whether the instinct was right.
Discussion prompt
A rectangle is 3 cm by 4 cm. Every length is doubled. What happens to its area?
Hint: Compute both areas rather than reasoning about the factor.
Answer:
The area is multiplied by 4, not by 2. The original area is 12 square cm; the new rectangle is 6 by 8, with an area of 48 square cm, and 48 divided by 12 is 4.
Why: area is a product of two lengths, and both of them doubled. Two doublings multiply by 2 times 2.
The intuitive answer, that the area also doubles, is what almost everyone says first, and it is always among the choices.
The general rule follows immediately: multiply every length by k and the area is multiplied by k squared. For a volume, three lengths scale, so it is multiplied by k cubed.
Pattern
Three steps, and the first one is looking something up.
Open the reference sheet and copy out the formula you need.
Why: It is provided, it is one tap away, and copying it removes any risk of misremembering. There is no credit for working from memory.
Substitute the numbers, watching the radius-versus-diameter distinction.
Why: Circle formulas take the radius. A question that gives you a diameter is expecting you to halve it, and it will offer the un-halved answer.
Attach the units and check they match what was asked.
Why: Square units for an area, cubic for a volume. If the units are wrong, the formula was wrong, and you know before checking the answer.
The units check in step 3 is not decoration. It catches a wrong-formula error in about two seconds, before the answer is selected.
Section
Section 2
Concept
Every geometry formula the SAT expects is one tap away inside the app for the whole test.
| given to you | not given — you must know it |
|---|---|
| Area and circumference of a circle | Percent change as a multiplier |
| Area of a triangle and a rectangle | Slope from two points |
| The Pythagorean theorem | The quadratic formula |
| 30-60-90 and 45-45-90 triangles | SOH-CAH-TOA |
| Volume of box, cylinder, sphere, cone, pyramid | The exponent rules |
| 360 degrees and 2 pi radians in a circle | The scaling rule for area and volume |
Time spent memorising the left column is time taken from the right column. Spend it on the algebra instead.
Concept
Length is linear, area is square, volume is cubic — and the units come out of the formula automatically.
Treat the units as a free check that runs alongside the arithmetic rather than as bookkeeping at the end.
Prediction
The rule that beats intuition.
Predict first
A triangle's base and height are both tripled. Its area is multiplied by what?
Correct: 9
Why: Area multiplies two lengths, and both were tripled, so the area is multiplied by 3 times 3, which is 9. Checking with numbers: a base of 2 and height of 4 gives an area of 4, while a base of 6 and height of 12 gives 36, and 36 divided by 4 is 9. The answer 3 is the intuitive one and is wrong; 27 would be the volume answer, where three lengths scale.
Concept
Multiply every length by k, and lengths scale by k, areas by k squared, and volumes by k cubed.
| if every length is | perimeter becomes | area becomes | volume becomes |
|---|---|---|---|
| doubled (k = 2) | 2 times | 4 times | 8 times |
| tripled (k = 3) | 3 times | 9 times | 27 times |
| halved (k = one half) | half | a quarter | an eighth |
This is the single most valuable fact in the deck, and it is the one geometry fact the reference sheet does not give you.
Prediction
Halve it before substituting.
Predict first
A circle has diameter 10. What is its area?
Correct: 25 pi
Why: The radius is half the diameter, so r equals 5, and the area is pi times 5 squared, which is 25 pi. Using the diameter as the radius gives 100 pi — four times too large, because the error is inside a square. That is why this trap is more costly for areas than for circumferences, where it only doubles the answer.
Concept
Break an awkward figure into standard shapes, then add their areas or subtract the hole.
The phrase shaded region is a reliable signal for subtraction. Find the two whole shapes and take one from the other.
Concept
Both the area and circumference formulas use r, so halve any diameter before substituting.
Circle the word radius or diameter in the stem the moment you see it. It is the highest-frequency single-word trap in the whole type.
Prediction
The units follow from the formula.
Predict first
A box measures 3 m by 4 m by 5 m. Its volume is 60 what?
Correct: cubic metres
Why: Volume multiplies three lengths together, so three sets of metres combine into cubic metres. Square metres would be an area, produced by multiplying only two lengths, and plain metres would be a single length such as an edge or a perimeter. The units are not an afterthought; they are produced by the formula itself.
Concept
Take the angle over 360, then multiply by the whole area or the whole circumference.
That last point catches people: perimeter of a sector includes the two straight sides, while arc length does not.
Concept
When the total is given and a dimension is missing, substitute what you know and solve.
The negative root is always discarded here for a physical reason rather than a mathematical one: a length cannot be negative.
Prediction
Angle over 360.
Predict first
A circle has area 36 pi. What is the area of its 60-degree sector?
Correct: 6 pi
Why: Sixty degrees is 60 over 360 of the circle, which is one sixth. One sixth of 36 pi is 6 pi. The most common error is dividing by 60 rather than taking 60 out of 360, and the second most common is forgetting to convert the angle into a fraction at all.
Two truths and a lie
Three of these geometry statements are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: c
Why: Doubling the radius multiplies the area by four, not two, because the radius appears squared in the area formula. A circle of radius 3 has area 9 pi; radius 6 gives 36 pi. This is the scaling rule in its most commonly tested single instance, and the intuitive answer of doubling is offered on essentially every version of the question.
Check
Apply the rule rather than the instinct.
Check your understanding
A rectangular prism has volume 24 cubic cm. If every dimension is tripled, what is the new volume?
Answer: A
Why: Every length is multiplied by 3, so the volume is multiplied by 3 cubed, which is 27. Then 24 times 27 is 648. Checking with a concrete prism: 2 by 3 by 4 gives 24, and 6 by 9 by 12 gives 648.
Choice B is the answer almost everyone gives before learning the rule, which is exactly why this question is worth a mark to anyone who has.
Section
Section 3
Worked example
A cylindrical tank has radius 4 m and height 5 m. What is its volume, in cubic metres, in terms of pi?
Figure (svg): A cylinder with its radius and height labelled
Copy the formula from the reference sheet: volume equals pi times r squared times h.
Why: It is provided, so there is nothing to recall and nothing to misremember.
Substitute r equals 4 and h equals 5: pi times 16 times 5.
Why: Square the radius before multiplying by the height, since only r is squared.
Multiply: 80 pi cubic metres.
Why: Sixteen times five is eighty, and pi is left in the answer as the question requested.
The wrong answer to watch for is 400 pi, which comes from squaring the height as well, or from using 20 as the radius.
Verify: check the units: metres times metres times metres gives cubic metres.
Why: Three lengths were multiplied, which is correct for a volume.
Answer: 80 pi cubic metres
Worked example
A cone has volume 30 cubic cm. Every dimension is doubled. What is the new volume?
Figure (svg): Bars contrasting doubling the volume with multiplying it by eight
Identify the scale factor: every length is multiplied by k equals 2.
Why: The rule needs the factor applied to lengths, which is what doubling every dimension means.
Volume scales by k cubed, which is 2 cubed, or 8.
Why: A volume is a product of three lengths, so each of the three doublings contributes a factor of 2.
So the new volume is 30 times 8, which is 240 cubic cm.
Why: The original volume is multiplied by the scale factor for volume.
Notice that you never needed the cone's dimensions or even its formula. The scaling rule works from the volume alone.
Verify: check with a concrete cone: radius 3 and height 10 gives 30 pi, and radius 6 and height 20 gives 240 pi.
Why: The ratio is 8, confirming the rule independently of the particular shape.
Answer: 240 cubic cm
Worked example
A circle of radius 5 sits inside a square of side 10, touching all four sides. What is the shaded area outside the circle but inside the square, in terms of pi?
Figure (svg): A circle of radius five inscribed so that it touches each side of a square
Area of the square: side times side, so 10 times 10, which is 100.
Why: The circle touching all four sides means its diameter equals the side length.
Area of the circle: pi times 5 squared, which is 25 pi.
Why: The radius is 5, given directly, so no halving is needed here.
Subtract: the shaded area is 100 minus 25 pi.
Why: Shaded region means the outer shape with the inner one removed.
Answers to shaded-region questions are usually left in terms of pi precisely because the subtraction does not simplify. An exact expression is the expected form.
Verify: check the size: 25 pi is about 78.5, so the shaded area is about 21.5, comfortably less than the square.
Why: A plausible fraction of the square is left over, which a sign error would not produce.
Answer: 100 - 25 pi square units
Step zero
Before substituting anything.
Discussion prompt
A question gives the diameter of a circular pond as 14 m and asks for its area. What is the very first thing you write, and what is the trap?
Hint: What does the area formula actually take?
Answer:
You write r equals 7, because the formula takes the radius and you were given the diameter.
Then area equals pi times 7 squared, which is 49 pi square metres.
The trap is substituting 14, giving 196 pi. That is four times too large, because the error sits inside a square.
Why it is worse for area than for circumference: using the diameter in the circumference formula doubles the answer, but using it in the area formula quadruples it.
The habit: the moment you read the word diameter, halve it and write the radius down. Do not carry the diameter into the substitution.
Worked example
A circle has radius 6. What is the area of a 120-degree sector, and what is that sector's perimeter?
Figure (svg): A 120-degree sector of a circle of radius six
The fraction is 120 over 360, which is one third.
Why: A sector is that fraction of the whole circle for both area and arc.
Area: one third of pi times 36, which is 12 pi.
Why: The whole circle has area 36 pi, and the sector takes a third of it.
Arc length: one third of the circumference 12 pi, which is 4 pi. Perimeter adds the two radii: 4 pi plus 12.
Why: The boundary of a sector is the arc plus the two straight edges.
Arc length and perimeter are different quantities, and the question will say which it wants. Forgetting the two radii is the standard error.
Verify: check that the arc, about 12.6, is less than the perimeter, about 24.6.
Why: The two radii add 12, so the perimeter must exceed the arc by exactly that.
Answer: area 12 pi; perimeter 4 pi + 12
Worked example
A rectangular box has a square base, a height of 6 cm, and a volume of 150 cubic cm. What is the side of the base?
Figure (svg): A box with a square base of side s and height six
Let the base side be s. Then the volume is s times s times 6, which is 6 s squared.
Why: A square base means length and depth are the same unknown.
Set that equal to 150: 6 s squared equals 150, so s squared equals 25.
Why: Divide by the known height first, leaving only the squared unknown.
Take the positive square root: s equals 5 cm.
Why: The negative root is discarded because a length cannot be negative.
Divide by everything known before taking a root. Taking the root too early is where the arithmetic goes wrong.
Verify: check forwards: 5 times 5 times 6 is 150 cubic cm.
Why: Substituting the answer into the original formula reproduces the given volume.
Answer: s = 5 cm
Faded example
From memory. The one geometry fact not on the reference sheet.
Fill in the blanks
If every length is multiplied by k, then lengths scale by k, areas scale by k squared, and volumes scale by k cubed. So doubling every dimension multiplies a volume by 8.
Why: The reason is worth carrying rather than the rule alone: an area is a product of two lengths and a volume of three, so each scaling is applied that many times. Once you have the reason, you can reconstruct the rule under pressure even if the exponents momentarily desert you.
Worked example
A cube has edge 4 cm. Find its volume and its surface area, and say which units each takes.
Figure (svg): A table comparing the volume and surface area of a cube of edge four
Volume: edge cubed, so 4 times 4 times 4, which is 64 cubic cm.
Why: Three lengths multiplied gives cubic units.
One face has area 4 squared, which is 16 square cm.
Why: A face of a cube is a square of side equal to the edge.
A cube has 6 faces, so the surface area is 6 times 16, which is 96 square cm.
Why: Surface area totals the outside faces, and it measures flat regions.
That the surface area number exceeds the volume number here means nothing at all. They are different quantities and cannot be compared.
Verify: check the units differ: 64 is cubic and 96 is square.
Why: The two numbers are close in size but measure completely different things, which the units make explicit.
Answer: volume 64 cubic cm; surface area 96 square cm
Fill the middle
Both formulas take the radius.
Fill in the blanks
The area of a circle is pi times r squared. Its circumference is 2 pi r. If you are given the diameter, first halve it. And a 90-degree sector is one quarter of the whole circle.
Why: Both circle formulas take r, which is exactly why the diameter trap works so reliably. Writing the radius down as a separate line before substituting costs two seconds and removes the error entirely.
Estimation
Pi is a little over 3, which is enough for a sanity check.
Predict first
A circle has radius 10. Roughly what is its area?
Correct: about 314
Why: The area is pi times 100, and pi is a little over 3, so the answer is a little over 300. The exact value is about 314.16. The answer 63 is the circumference, 2 pi times 10, which is about 62.8 — a genuinely computed quantity, and the wrong one. The answer 100 is r squared with pi forgotten entirely.
Check
Two traps in one question.
Check your understanding
A circular garden has diameter 8 m. What is its area?
Answer: A
Why: The radius is half the diameter, so r equals 4, and the area is pi times 4 squared, which is 16 pi square metres. Area multiplies two lengths, so the units are square metres.
Three of the four choices are correct answers to adjacent questions. The units check alone eliminates D, and writing the radius down eliminates B.
Section
Section 4
Trap
The trap. Every dimension of a container is doubled, and you conclude that it now holds twice as much.
It holds eight times as much. Volume is a product of three lengths, and all three doubled.
The same error in two dimensions says a doubled rectangle has twice the area, when it has four times.
The fix. Ask how many lengths are multiplied together in the quantity you are scaling.
One length means k, two means k squared, three means k cubed.
Trap
The trap. The stem gives a diameter of 12 and you substitute 12 into pi r squared, getting 144 pi.
The radius is 6, so the area is 36 pi. Your answer is four times too large.
The error is invisible in the working because the arithmetic is flawless — only the input was wrong.
The fix. Both circle formulas take the radius, so convert before you substitute.
Write r equals half the diameter as its own line, then substitute from that line.
Error analysis
A student's reasoning about a scaled container. One inference is wrong.
Annotate
On: \( \text{all lengths} \times 3 \;\Rightarrow\; \text{volume} \times 3 \)
When a scaling rule is hard to recall under pressure, build a 1 by 1 example and scale it. It reconstructs the rule in about ten seconds and cannot be misremembered.
Trap
The trap. A question asks how much wrapping paper covers a box, and you compute its volume.
Wrapping covers the outside, which is surface area, measured in square units. Volume measures what fits inside.
The two are frequently close in size for small solids, so the number alone does not look wrong.
The fix. Read the verb. Covers, wraps, paints and tiles all mean surface area; fills, holds and contains mean volume.
Then confirm with the units: square for area, cubic for volume.
Elimination
A sphere's radius is halved.
Eliminate the wrong options
What happens to its volume? Three choices can be ruled out by the scaling rule alone.
Survives elimination: b
Why: The scale factor is one half, and a volume scales by k cubed, so the new volume is one half cubed, which is one eighth. Notice that the three wrong answers are the three adjacent powers: k to the zero, k to the one, and k squared. Recognising the pattern of distractors is itself a shortcut, because the correct answer on a scaling question is nearly always the one with the highest power.
Trap
The trap. Asked for the perimeter of a sector, you give the arc length alone.
A sector is bounded by the arc and two radii. Its perimeter is the arc plus twice the radius.
The arc-only answer is always offered, and it is the answer to a different, adjacent question.
The fix. Trace the boundary of the shape with your finger and count every edge you cross.
For a sector that is one curved edge and two straight ones.
Counterexample
A claim that sounds like it should follow.
Discussion prompt
A student says: if two rectangles have the same perimeter, they have the same area. Give a counterexample.
Hint: Try a long thin rectangle against a square.
Answer:
Counterexample: a 1 by 9 rectangle and a 5 by 5 square. Both have perimeter 20, but the areas are 9 and 25.
The mechanism: perimeter adds the lengths while area multiplies them, and a fixed sum can be split in many ways with very different products.
The extreme case makes it clearest: a 0.5 by 9.5 rectangle also has perimeter 20 and an area of only 4.75.
The general fact: for a fixed perimeter, the area is largest when the shape is most compact — a square among rectangles, and a circle among all shapes.
The SAT tests this by offering shapes with matching perimeters and asking which has the greater area, and the answer is always the more square-like one.
Edge cases
The rule assumes something that questions sometimes break.
Discussion prompt
The scaling rule says areas grow by k squared. What must be true for it to apply, and what happens when only one dimension changes?
Hint: The rule says every length.
Answer:
The rule requires that EVERY length is multiplied by the same k — the shape stays the same and only its size changes.
If only one dimension changes, the rule does not apply. Doubling only the height of a rectangle doubles its area, because only one of the two multiplied lengths changed.
Doubling only the height of a cylinder doubles its volume; doubling only its radius multiplies the volume by four, because the radius appears squared.
So read the stem carefully: every dimension doubled means k cubed for volume, while the height is doubled means a factor of two.
The reliable general method when only some dimensions change: look at the formula and apply each factor to the letters it actually affects. The k-squared and k-cubed shortcuts are the special case where everything scales together.
Check
Rearrange rather than substitute forwards.
Check your understanding
A cylinder has volume 96 pi cubic cm and height 6 cm. What is its radius?
Answer: A
Why: Substitute into pi r squared h: 96 pi equals pi times r squared times 6. Dividing both sides by pi gives 96 equals 6 r squared, so r squared equals 16 and r equals 4. Checking forwards: pi times 16 times 6 is 96 pi.
Choice B is the standard incompleteness on this variant: the algebra reached r squared and stopped one operation early.
Section
Section 5
Matching
Six situations, six formulas. No arithmetic.
Match the pairs
Why: Notice how reliably the verb decides the formula: around gives a perimeter or circumference, covers gives an area, fills gives a volume. Four of these six could be answered from the verb alone, without reading the rest of the sentence.
Sorting
Every length is multiplied by k. What happens to the stated quantity?
Sort into buckets
By what factor does it change?
The shape is irrelevant — square, circle or sphere all follow the same rule, because scaling leaves the constant in the formula untouched and only affects the lengths.
Comparison
Fill the blanks from memory.
Comparison matrix
| measure | lengths multiplied | units | scales by |
|---|---|---|---|
| Perimeter | one | linear, such as cm | k |
| Area | two | square, such as cm squared | k squared |
| Surface area | two | square, such as cm squared | k squared |
| Volume | three | cubic, such as cm cubed | k cubed |
Every column follows from the first: how many lengths are multiplied decides the units and the scaling power together. That is why the units check catches a scaling error as well as a formula error.
Trade off
Fill in what each is worth on this type.
Comparison matrix
| what you could learn | questions it affects | worth the time? |
|---|---|---|
| The volume formulas | none — they are provided | no, it is given to you |
| The scaling rule | about one per test | yes, it is not provided |
| Radius versus diameter | most circle questions | yes, it is a habit not a fact |
| Where the reference sheet is | every geometry question | yes, and it takes one minute |
The top row is the point. The instinct to memorise formulas is exactly backwards on this type: the formulas are free, and the habits are what cost you marks.
Real world
One minute on why the scaling rule matters.
Discussion prompt
Why do large animals have proportionally thicker legs than small ones, and what has that got to do with k squared and k cubed?
Answer:
Because weight scales with volume and strength scales with area. An animal scaled up by k weighs k cubed times as much, but its bones are only k squared times as strong, since strength depends on cross-sectional area.
So doubling an animal's size makes it eight times heavier while its legs are only four times stronger. The legs must get proportionally thicker to compensate.
The same rule explains why small creatures survive falls — surface area, which creates drag, has not shrunk as fast as mass.
And it explains cooking times: heat enters through the surface, area k squared, but there is volume k cubed to heat, so a doubled roast takes far more than twice as long.
The SAT version is a plain question about a scaled container, but it is the same fact, and knowing why it holds makes it impossible to forget.
Ranking
All four have perimeter 24. Order them from smallest area to largest.
Put in order
Why: The areas are 11, 20, 32 and 36. All four have the same perimeter of 24, and yet the largest area is more than three times the smallest. The pattern is that the more square-like the rectangle, the greater its area for a fixed perimeter — which is why the 6 by 6 square wins. This is the counterexample slide made quantitative, and it is exactly what a which-has-the-greater-area question is testing.
Concept
This type is 5.3 per cent of the section. Since the formulas are provided, almost all the available gain is in three habits.
| session | what you do | why |
|---|---|---|
| 1 | Open the reference sheet and work fifteen direct-substitution questions from it, never from memory. | Builds the habit of looking, and shows how little recall is actually needed. |
| 2 | Ten scaling questions, stating k and the power before computing. | The one rule not provided, and the one most students get wrong. |
| 3 | Ten circle questions, writing the radius on its own line every time. | Kills the diameter trap, which is the highest-frequency error in the type. |
| 4 | Mixed set including composite figures and backwards questions, checking units on every answer. | Composite and backwards are the two variants students prepare least. |
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Explain it to yourself
Close the deck.
Discussion prompt
Without looking, state the scaling rule for lengths, areas and volumes, explain WHY each power is what it is, and give the doubling case in numbers.
Hint: Count how many lengths each quantity multiplies.
Answer:
Lengths scale by k, because a length is one length.
Areas scale by k squared, because an area multiplies two lengths and both of them scaled.
Volumes scale by k cubed, because a volume multiplies three lengths and all three scaled.
Doubling: perimeter times 2, area times 4, volume times 8.
If you gave the reason as well as the rule, you can rebuild it under pressure — and the reason also tells you what to do when only one dimension changes.
Explain it
Two minutes, out loud.
Discussion prompt
A friend says that doubling a box's dimensions doubles what it holds. How do you convince them, without asserting a rule?
Answer:
Do not state the rule — build the smallest example. Take a 1 by 1 by 1 cube. It holds 1 cubic unit.
Now double every edge: a 2 by 2 by 2 cube. Ask them to count: it holds 8 cubic units, and you can literally see eight small cubes inside it.
Then ask why: because the box got twice as long, twice as wide AND twice as tall. Three separate doublings, so 2 times 2 times 2.
Extend it to area so the pattern is visible: a 1 by 1 square doubled becomes 2 by 2, holding four unit squares, because two lengths doubled.
Then let them state the rule themselves. A rule they reconstructed from a picture survives test-day pressure; one they were told does not.
Commit first
Commit before you check.
Predict first
A cylinder's radius and height are both halved. Its volume becomes what fraction of the original?
Correct: one eighth
Why: Every length is multiplied by one half, so the volume is multiplied by one half cubed, which is one eighth. Checking directly: pi r squared h with both r and h halved gives pi times r squared over 4 times h over 2, which is the original over 8. Note that if only the radius were halved, the answer would be one quarter — because the radius appears squared and the height does not.
Connect it up
Blank paper.
Draw it
Draw a map of area and volume. Down the left, list the three measures: perimeter, area, volume. Beside each write how many lengths it multiplies, its units, and its scaling power, so the three columns line up as 1, 2, 3. In the middle, draw a 1 by 1 square next to a 2 by 2 square, and a 1 by 1 by 1 cube next to a 2 by 2 by 2 cube, with the counts 1, 4, 1 and 8 written inside. On the right, list the four traps: scaling by the wrong power, using the diameter as the radius, confusing surface area with volume, and giving an arc length when a sector perimeter was asked for. At the bottom, write in large letters: THE FORMULAS ARE PROVIDED.
Exit ticket
One question before you close the deck.
Predict first
A geometry question needs the volume of a cone and you cannot recall the formula. What do you do?
Correct: Open the reference sheet, which has it
Why: The reference sheet is available for the entire test and contains every volume formula the SAT uses. Not remembering it costs nothing at all. Deriving it wastes a minute on something already provided, guessing throws away a question you can certainly answer, and skipping is only sensible when a question is genuinely beyond you — which this is not.
Recap
One type, three habits: look it up, watch the units, and know how things scale.
| never do this | do this instead |
|---|---|
| Say a doubled solid holds twice as much | Cube the scale factor: it holds eight times as much |
| Substitute the diameter into pi r squared | Halve it first and write the radius down |
| Compute a volume when asked what covers a surface | Read the verb: covers means area, fills means volume |
| Give the arc length as a sector's perimeter | Add the two radii |
| Memorise the volume formulas | Open the reference sheet and spend the time on algebra |
| Stop at r squared | Take the square root, and keep the positive value |
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