Understanding Similar Cubes and Capacity

When we talk about similar cubes, we mean that they have the same shape but different sizes. In geometry, similar figures are figures that are proportional to each other. This means that the ratio of any two corresponding lengths in the figures is constant. This constant ratio is often called the scale factor.

For similar three-dimensional figures like cubes, the relationship between corresponding lengths, areas, and volumes (or capacity) is specific:

• The ratio of corresponding lengths is the scale factor.
• The ratio of corresponding areas (like surface area) is the square of the scale factor.
• The ratio of corresponding volumes (or capacity) is the cube of the scale factor.

Calculating the Scale Factor from Heights

We are given two similar cubes with heights of 8 cm and 12 cm. The height of a cube is the same as its side length. Let's find the scale factor, which is the ratio of their corresponding heights.

Ratio of heights = \(\frac{\text{Height of smaller cube}}{\text{Height of bigger cube}}\)

Ratio of heights = \(\frac{8 \text{ cm}}{12 \text{ cm}}\)

Simplifying the fraction, we get:

Scale Factor = \(\frac{8}{12} = \frac{2}{3}\)

So, the scale factor from the smaller cube to the bigger cube is \(2/3\). This means every length in the smaller cube is \(2/3\) times the corresponding length in the bigger cube.

Relating Height Ratio to Capacity Ratio

As mentioned, for similar figures, the ratio of their capacities (volumes) is the cube of the scale factor (ratio of corresponding lengths).

Ratio of Capacities = (Ratio of heights)³

Ratio of Capacities = \(\left(\frac{2}{3}\right)^3\)

Ratio of Capacities = \(\frac{2^3}{3^3} = \frac{8}{27}\)

This means the capacity of the smaller cube is \(8/27\) times the capacity of the bigger cube. We can write this as:

\(\frac{\text{Capacity of smaller cube}}{\text{Capacity of bigger cube}} = \frac{8}{27}\)

Finding the Capacity of the Bigger Cube

We are given that the capacity of the smaller cube is 80 cm³. We can use the ratio we just found to calculate the capacity of the bigger cube.

Let \(V_{smaller}\) be the capacity of the smaller cube and \(V_{bigger}\) be the capacity of the bigger cube.

\(\frac{V_{smaller}}{V_{bigger}} = \frac{8}{27}\)

Substitute the given value \(V_{smaller} = 80 \text{ cm}^3\):

\(\frac{80}{V_{bigger}} = \frac{8}{27}\)

Now, we can solve for \(V_{bigger}\) by cross-multiplying:

\(8 \times V_{bigger} = 80 \times 27\)

\(V_{bigger} = \frac{80 \times 27}{8}\)

\(V_{bigger} = 10 \times 27\)

\(V_{bigger} = 270\)

The capacity of the bigger cube is 270 cm³.

Summary of Steps

1. Identify the corresponding lengths and their ratio (the scale factor).
2. Cube the scale factor to find the ratio of the volumes (capacities).
3. Use the volume ratio and the known volume to set up a proportion.
4. Solve the proportion to find the unknown volume.

In this case:

• Ratio of heights = \(8/12 = 2/3\).
• Ratio of capacities = \((2/3)^3 = 8/27\).
• \(\frac{80}{V_{bigger}} = \frac{8}{27}\)
• \(V_{bigger} = \frac{80 \times 27}{8} = 270\) cm³.

Revision Table: Similar Figures Ratios

Additional Information on Similar Geometric Figures

Similar geometric figures maintain proportional relationships between all their corresponding parts. This principle applies not just to cubes, but to any similar polygons or polyhedra. For example, similar triangles, similar spheres, or similar cylinders follow the same ratio rules regarding lengths, areas, and volumes.

• The scale factor (\(k\)) is usually defined as the ratio of a length in the new or larger figure to the corresponding length in the original or smaller figure. If we had taken the ratio of heights as \(12/8 = 3/2\) (bigger to smaller), then \(k = 3/2\). In that case, the ratio of capacities (bigger to smaller) would be \((3/2)^3 = 27/8\). Using this, \(\frac{V_{bigger}}{V_{smaller}} = \frac{27}{8}\), so \(V_{bigger} = \frac{27}{8} \times V_{smaller} = \frac{27}{8} \times 80 = 27 \times 10 = 270\) cm³. The result is the same, regardless of which figure is considered 'smaller' or 'bigger' for calculating the initial ratio, as long as consistency is maintained.
• Understanding the relationship between linear dimensions, area, and volume ratios is fundamental in geometry and scaling problems. It allows us to calculate properties of scaled objects without needing all their dimensions, provided we know the properties of the original object and the scale factor.
• Capacity is often used interchangeably with volume, especially when referring to the amount a container can hold. For a cube, the volume is calculated as side length cubed (\(s^3\)). Since the heights are the side lengths of these cubes, the principle of similar solids applies directly to their volumes or capacities.