相似図形 ⟦number⟧

The architect's blueprint

• An architect draws a building at 1:100 scale. Every measurement on the plan is $\dfrac{1}{100}$ of the real thing.
• The drawing and the building are similar 相似 — same shape, different size.

What makes shapes similar?

• Two shapes are similar if they have the same angles and all sides are multiplied by the same scale factor 比例因子 $k$.
• All circles are similar. All equilateral triangles are similar. Not all rectangles are similar.
• Shapes that are the exact same size and shape are called congruent 全等.

The three angles of a triangle add up to 180 degrees

Area and volume scale differently (Extended) (Extended)

• Lengths scale by $k$, but:

Worked example

• Two similar solids have lengths in the ratio $2:3$.
• Area ratio $= 2^2:3^2 = 4:9$.
• Volume ratio $= 2^3:3^3 = 8:27$.
• If the smaller has volume $40$ cm³: larger $= 40 \times \dfrac{27}{8} = 135$ cm³.

Fun fact

• The surface area of a sphere scales as $r^2$, but its volume scales as $r^3$. That's why elephants need big ears (surface area for cooling) while mice don't — volume grows faster than surface area.

Two squares are always similar — same angles, sides scaled by a factor. Here the scale factor is $2$, so the area ratio is $4$.

Find a missing corresponding length

• Corresponding sides 4 cm and 10 cm give $k=10/4=2.5$ from small to large. A 6 cm side therefore becomes $L=kl=2.5(6)=15$ cm.
• In reverse, a 20 cm large side comes from $l=L/k=20/2.5=8$ cm. Core uses length ratios; Extended also uses area and volume ratios.

You've got it