IMPACT OF SCALING ON AREA AND VOLUME

Impact of Scaling on Area and Volume

Impact of Scaling on Area and Volume

Scaling is used in science to determine the effect of size on function. Can we live normal, active lives if reduced to the size of an ant or if increased to the size of King Kong? Scaling is also used to construct models of larger or smaller objects. For example, a geologist can build a model whose changes in a short time represent structural changes in the earth that take millions of years to occur. Engineers and architects use scaling to make models for testing designs of buildings, airplane wings, power plants and so forth.

To start our discussion of scaling, consider two groups of blocks. The first group consists of two cubes one sitting on the other. Each cube has sides of length 1 unit. The two blocks together have dimensions 1x 1 x 2.

The second group of blocks has the same shape as the first but is three times longer in each direction. Its dimensions are 3 x 3 x 6. The scaling factor of the second group of blocks compared to the first is 3 since each dimension of the second group is three times longer than the similar dimension of the first group. The scaling factor ? is the ratio of a dimension d' of one object and the similar dimension d of another object whose shape is the same as the first.

If the objects have similar shapes, any dimension can be used to calculate the scaling factor, just as long as the same dimension is used for each object.

How does the area of a similarly shaped objects depend on the scaling factor? Consider the cross section area of the first group of blocks described earlier. The cross-section area A = 1 x 1 = 1, whereas for the larger set of blocks, the cross-section are A' = 3 x 3 = 9. The ratio of the two areas A'/A = 9/1 = 9 is the square of the scaling factor. The ratio of the areas of the side surfaces of the blocks also equals the scaling factor squared since , for the side surfaces A'/A = (3 x 6)/(1 x 2 ) = 0.

Areas scale as the square of the scaling factor

This seems reasonable. Each dimension has increased by ? , and the area is the product of two dimension. How do the volumes of these two similarly shaped groups of blocks compare? The volume of the small group of blocks is 1 x 1 x 2 = 2, whereas that of the larger groups is 3 x 3 x 6 = 54. The ratio of their volumes V'/V = 54/2 = 27, is the cube of the scaling factor (33 = 27). This also seems reasonable. Each dimension has increased by ? , and the volume is the product of three dimensions. Thus, the volume of one object should be times that of the other ...