Math CentralQuandaries & Queries


Question from gina, a parent:

2 rectangular prisms are similar by a scale of 0.75. The volume of the smaller prism is 27cm3 (cubed). How many cubic centimeters is the volume of the larger prism?

Hi Gina,

When the statement is that "the scale is 0.75" it means that the linear scale is 0.75. Since $0.75 = \large \frac34$ the statement that "the scale is 0.75" means that

\[\mbox{(a linear measurement of the smaller prism)}= \frac34 \times \mbox{(the same linear measurement on the larger prism.)}\]

Multiplying both sides of this equation by $\large \frac43$ gives

\[\mbox{(a linear measurement of the larger prism)}= \frac43 \times \mbox{(the same linear measurement on the smaller prism.)}\]

Suppose what you have is a right rectangular prism so its area is

\[\mbox{length} \times\mbox{ width} \times \mbox{ height.}\]

Suppose the length, width and height of the larger prism are $L, W$ and $H$ and for the smaller prism are $l, w$ and $h.$ Then

\[L = \frac43 l, W = \frac43 w \mbox{ and } H = \frac43 h\]

and thus

\[L \times W \times H = \frac43 l \times \frac43 w \times \frac43 h = \left(\frac43 \right)^3 l \times w \times h.\]

Thus the volume of the larger prism is $\large \left(\frac43 \right)^3$ times the volume of the smaller prism.

The factor of the cube of the scaling factor applies in many situations. If you scale up a three dimensional object by a linear scale of $k$ then the volume of the scaled up object is $k^3$ times the volume of the smaller object.


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