L2 · The area scale factor: the square of the linear scale factor

Learning goal

“I can move between the linear scale factor and the area scale factor in both directions - squaring to get from lengths to areas, and taking the square root to get back - and I can find areas and corresponding lengths for similar figures.”

Linear scale factor — the Lesson L1 scale factor k: it multiplies LENGTHS.

Area scale factor — the multiplier for AREAS of similar figures: it equals k^2, the square of the linear scale factor.

Corresponding areas — the areas of two similar figures matched to each other, just as corresponding lengths are matched.

Worked example

A square has side 3 cm, so its area is 9 cm2. It is enlarged with linear scale factor k = 2. Step 1 (lengths): new side = 3 x 2 = 6 cm. Step 2 (areas, directly): new area = 6 x 6 = 36 cm2. Step 3 (the pattern): 36 / 9 = 4, and 4 = 2^2. The area multiplier is the SQUARE of the length multiplier - because area is built from two lengths, and each one took the factor k. Say it plainly: lengths scale by k; areas scale by k^2. Doubling a figure's lengths quadruples the material needed to cover it.

Check yourself — explain it

The area multiplier is k^2, not 2k, because...

Paired problem — example still visible

With k = 3, an original area of 5 cm2 becomes what? -> Area scale factor 3^2 = 9; 5 x 9 = 45 cm2.

Faded completion — last step is yours

"A reduction has k = 1/2. Area scale factor = (1/2)^2 = ______. An original area of 48 cm2 becomes 48 x ______ = ______ cm2." -> 1/4; 1/4; 12 cm2

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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