Make a shape twice as big and you get four of it.

Not twice as much. Four times as much.

Nobody believes that from a formula, so here it is counted out.

Count the copies.

Double every side of a square and four of the original fit inside it.

Triple them and nine fit.

Sides doubled, four copies. Sides tripled, nine. The numbers on the cells are there to be counted.

It is not a square thing. A triangle cuts into 4 and then 9 the same way.

Because area is a length times a length.

Scale the figure and both of those lengths pick up the same k.

So the k comes out twice. That is the entire k².

Two lengths, one scale factor, and it lands on the area twice.

Move it yourself.

k is the scale factor. Drag it and watch the three bars come apart.

Scale factor k = 2

every length ×k: ×2
the surface area ×k²: ×4
the volume ×k³: ×8

Add a third direction and the k lands a third time.

Double every edge of a cube and eight of the original fit inside.

Not four. Eight.

Pulled apart so all eight are visible. Solid, you would only ever see three of them.

Everything above, in one place

×k
Every length: ×k²
Every area — surface, cross-section, the lot: ×k³
Every volume

One condition, and it is the whole condition: the two figures have to be similar. Same shape, every length scaled by the same k.

This has nothing to do with straight edges.

Cover any outline you like with grid squares.

Doubling turns every one of those squares into four.

And the shape is nothing but its squares.

Follow the dark cell. One square in, four squares out — and that happens to every square at once.

Outlines nobody would call a rectangle, each drawn at double size around a copy of its old self. Area ×4 every time.

On the test they hand you the area and want the side.

So you go back down the ladder instead of up it.

Areas 9 to 25 means sides 3 to 5. Root it, don't halve it.

Both routes start at the lengths. You never cube an area.

Three ways this gets you.

“20% bigger” is 44% more area.

1.2 squared is 1.44, and the extra 0.44 is where the answer choice you wanted lives.

A 10 by 10 grown 20% is 12 by 12. The dark strip is 44 cells.

One side moved, so the rule is switched off.

Stretch a rectangle in one direction and the new shape is not similar to the old one. No k, so no k².

Width doubled, height untouched. Area doubled, and that is all.

Surface and volume do not grow together.

Volume runs away faster, so surface per unit of volume drops by a factor of k. Big things have less skin for their size.

Edge doubled: surface times four, volume times eight, ratio cut in half.

Five to try.

Do the move in your head first, then open it.

The last one is not a scaling problem. That is the point of it.

Triangle ABC is similar to triangle DEF, and AB : DE = 3 : 5. The area of ABC is 18. What is the area of DEF?
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50.
Sides run 3 to 5, so areas run 9 to 25. 18 × 25/9 = 50.
A cone's radius and height are each multiplied by 4. Its volume was 15 cubic inches. What is the new volume?
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960 cubic inches.
Every length ×4, so volume ×4³ = 64. 15 × 64 = 960.
The radius of a circle is increased by 20%. By what percent does the area increase?
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44%.
Lengths ×1.2, so area ×1.2² = 1.44. That is 44% more, not 20%.
Two similar solids have volumes 27 and 125. The smaller has surface area 45. What is the surface area of the larger?
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125.
Cube-root the volumes: lengths run 3 to 5. So areas run 9 to 25, and 45 × 25/9 = 125.
A rectangle is 8 by 3. Its width is doubled and its height is left alone. By what factor does the area change?
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2, not 4.
Only one length moved, so the new rectangle is not similar to the old one and k² has nothing to act on.

There are about forty thousand more questions where these came from.

Ours sit in Study Hall with a tutor built in, so a miss like the last one turns into a lesson on the spot.

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