Similarity Calculations

• The Area Scale Factor is $k^2$. This applies to any 2D area, cross-sectional area, or 3D total surface area.
• The Volume Scale Factor is $k^3$. This applies to volume, mass (if density is uniform), and Capacity.

Worked Example: Finding Area from Lengths

A small monster toy has a height of $20$ cm and a surface area of $300$ cm$^2$. A mathematically similar large monster toy has a height of $120$ cm. Calculate the surface area of the large monster toy.

Step 1: Calculate the linear scale factor ($k$).

• $k = 120 \div 20 = 6$

Step 2: Calculate the Area Scale Factor ($k^2$).

• $k^2 = 6^2 = 36$

Step 3: Multiply the small area by the Area Scale Factor.

• $\text{Large Area} = 300 \times 36 = 10{,}800$ cm$^2$

Worked Example: Finding Length from Volumes

Two mathematically similar cones have volumes of $20$ cm$^3$ and $540$ cm$^3$. The height of the larger cone is $15$ cm. Calculate the height of the smaller cone.

Step 1: Calculate the Volume Scale Factor ($k^3$).

• $k^3 = 540 \div 20 = 27$

Step 2: Find the linear scale factor ($k$) by cube rooting.

• $k = \sqrt[3]{27} = 3$

Step 3: Divide the larger height by $k$ to find the smaller height.

• $\text{Small Height} = 15 \div 3 = 5$ cm

Transitioning Dimensions (Higher Tier)

• You cannot convert directly between area and volume scale factors.
• You must always find the linear scale factor ($k$) first by square rooting the Area Scale Factor or cube rooting the Volume Scale Factor.
• For example, if the volumes of two similar solids are in the ratio $27:64$, their lengths are in the ratio $\sqrt[3]{27}:\sqrt[3]{64}$ which simplifies to $3:4$.
• Their surface areas would then be in the ratio $3^2:4^2$, giving $9:16$.

Negative Scale Factors (Higher Tier)

• A negative scale factor ($k < 0$) means the shape is inverted and rotated $180^\circ$ through a center of enlargement.
• Physical measurements like length, area, and volume cannot be negative.
• To find a missing length when $k$ is negative, multiply by the absolute value $|k|$.
• Because any number squared is positive, a negative linear scale factor still results in a positive Area Scale Factor ($k^2$).