Similar Solids

What are Similar Solids?

Two 3D solids are similar if they have the exact same shape but different sizes. This means that all of their corresponding linear dimensions (such as height, radius, width, or slant height) are proportional. The ratio of these corresponding dimensions is called the scale factor, often represented by $k$.

The Rules of Scale Factors

When you scale a 3-dimensional object, its area and volume grow at different rates compared to its 1-dimensional length. If two similar solids have a linear scale factor of $k$, the following rules always apply:

• Linear Ratio (1D): The ratio of any corresponding lengths (heights, radii, perimeters) is $k$.
• Surface Area Ratio (2D): The ratio of their corresponding surface areas (or base areas) is $k^2$.
• Volume Ratio (3D): The ratio of their corresponding volumes is $k^3$.

For example, if you double the height and width of a box ($k = 2$), its surface area increases by $2^2 = 4$ times, and its volume increases by $2^3 = 8$ times.

Example Problems

Example 1: Finding the Volume of a Similar Solid Two similar cylinders have heights 6 and 9. If the smaller has a volume of $48\pi$, find the larger's volume.

Solution:

1. Find the linear scale factor $k$ going from the smaller cylinder to the larger one: $k = \frac{9}{6} = \frac{3}{2}$
2. Find the volume ratio. Since volumes scale by $k^3$, the ratio of the larger volume to the smaller volume is: $k^3 = \left(\frac{3}{2}\right)^3 = \frac{27}{8}$
3. Set up a proportion to find the larger volume ($V$): $\frac{V}{48\pi} = \frac{27}{8}$
4. Solve for $V$: $V = 48\pi \times \frac{27}{8} = 6\pi \times 27 = 162\pi$

The volume of the larger cylinder is $162\pi$.

Example 2: Finding Volume Ratio from Area Ratio Two similar cones have surface areas in the ratio $4:25$. Find the ratio of their volumes.

Solution:

1. The ratio of the surface areas is equal to $k^2$. Therefore: $k^2 = \frac{4}{25}$
2. Take the square root of both sides to find the linear scale factor $k$: $k = \sqrt{\frac{4}{25}} = \frac{2}{5}$
3. The ratio of the volumes is $k^3$. Cube the linear scale factor to find this ratio: $k^3 = \left(\frac{2}{5}\right)^3 = \frac{8}{125}$

The ratio of their volumes is $8:125$.