ARCH-M1-06: 2D/3D Geometric Similarity Scaling (k, k^2, k^3) & Frustums

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-M1-06: Why the 145s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~145s (Blows 90s Clock)

For mathematically similar 3D solids, the volume ratio is the cube of the linear ratio:

\frac{V_{\text{large}}}{V_{\text{small}}} = \left(\frac{4}{3}\right)^3 = \frac{64}{27}

. Since

108 \div 27 = 4

V_{\text{large}} = 4 \times 64 = 256\text{ cm}^3

(~95s).

The Fast ShortcutTarget: 35s (Saves 110s)

Line 1 (Cube the Ratio):

k = \frac{4}{3} \implies k^3 = \frac{64}{27}

.
**Line 2 (Cancel

27

):**

108 \times \frac{64}{27} = 4 \times 64 = 256\text{ cm}^3

(20s).

Where Students Lose Time: In 9 past UAT-UK questions (4% of Mathematics 1), candidates lose 110s by applying linear (k) or area (k²) scale factor to a 3d volume. Instead, apply M3 with Maths 1 skill M3.10.

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