Equivalent Solids

Two solids are considered equivalent when they have the same volume, even if their shapes are completely different. \[ V_A = V_B \] Here, \( V_A \) and \( V_B \) represent the volumes of the two solids.

This concept concerns only the amount of space a solid occupies - it has nothing to do with its shape or surface area.

Objects can look entirely different yet occupy the same volume.

So, equivalence doesn’t imply that solids share the same height, base, or surface area - only that they contain the same amount of space (volume).

A Practical Example

A cube with edges measuring 3 cm has a volume of 27 cm³:

\[ V = 3^3 = 27 \ cm^3 \]

A rectangular prism with a base of 3 cm × 3 cm and a height of 3 cm also has a volume of 27 cm³:

\[ V = 3^3 = 27 \ cm^3 \]

Likewise, a cylinder with a base area of 9 cm² and a height of 3 cm has the same volume:

\[ V = 9 \cdot 3 = 27 \ cm^3 \]

These are clearly different shapes, yet they are volumetrically equivalent.

Transformations That Preserve Volume

Many solids can be transformed into other shapes without altering their volume.

These transformations involve repositioning, cutting, and reassembling parts of the solid.

Example

A triangular prism can be divided into three pyramids of equal volume. By rearranging these pyramids, one can form a solid that is volumetrically equivalent to the original prism.

Note. Pyramids P1 and P2 share a congruent base $ ABC = A'B'C' $ and congruent height $ AA' = CC' $, making them equivalent. Pyramids P2 and P3 have congruent bases $ BCC' = BCB' $ and a common height $ A'B' $, so they too are equivalent. By the transitive property, P1 and P3 must also be equivalent.

The volume of a pyramid is given by the formula:

\[ V = \frac{1}{3} \cdot \text{base} \cdot \text{height} \]

While the volume of a prism is calculated as:

\[ V = \text{base} \cdot \text{height} \]

This shows that, when base and height are consistent, a pyramid has one-third the volume of a prism.

Therefore, combining three identical pyramids yields a prism of equal volume - proving that the two solids are equivalent.

Equivalence vs. Congruence

It's important to distinguish between equivalence and congruence.

• Congruence: solids have the same shape and size. For example, two identical cubes are congruent, and therefore also equivalent.
  
• Equivalence: solids share the same volume, regardless of shape. For example, a cube and a cylinder with equal volumes are equivalent, but not congruent.
  

And so on.