📐 Scaling Up! Crush Similar Solids’ Areas & Volumes 💥

Similarity isn’t just about shapes looking alike—it’s a **powerful shortcut** for finding surface areas and volumes without heavy computation. In O-Level exams you’ll often meet questions that scale a solid up or down and ask you for new measurements. Mastering this skill saves time and earns easy marks!

Understanding Similarity of Solids

For two similar 3-D figures, their corresponding lengths are in the same ratio, say

k

. Consequently, **area ratios become

k^{2}

** and **volume ratios become

k^{3}

**. Remember: *length → first power, area → square, volume → cube.*

If linear scale factor is $k$ , then $\dfrac{\text{Area}_2}{\text{Area}_1}=k^{2}$ .
If linear scale factor is $k$ , then $\dfrac{\text{Volume}_2}{\text{Volume}_1}=k^{3}$ .

Worked Example: Super-Sizing a Cube

Problem: A small cube has edge

4\,\text{cm}

. A similar larger cube has edge

10\,\text{cm}

. Find (i) the surface area and (ii) the volume of the larger cube.

Step 1: Identify linear scale factor
$k=\frac{10}{4}=2.5$
Step 2: Surface area of small cube
$A_1=6\times4^{2}=96\,\text{cm}^2$
Surface area factor $k^{2}=2.5^{2}=6.25$ so
$A_2=k^{2}A_1=6.25\times96=600\,\text{cm}^2$
Step 3: Volume of small cube
$V_1=4^{3}=64\,\text{cm}^3$
Volume factor $k^{3}=2.5^{3}=15.625$ so
$V_2=k^{3}V_1=15.625\times64=1000\,\text{cm}^3$

Practice scaling different solids—cylinders, cones, spheres—to gain confidence. **Consistent practice** will make these calculations second nature when exam time arrives!