Similarity in Shapes

This is different from congruent shapes, which have the same shape and the same size. Congruent shapes are a special case of similar shapes with length scale factor 1.

Before doing any calculation, identify which sides or edges correspond. Equal angles, distinctive corners and relative positions can help. It may also help to redraw rotated shapes in the same orientation.

Finding the length scale factor

Suppose a length on an original shape is changed into its corresponding length on a new shape. The length scale factor is

The direction matters. Reversing the journey gives the reciprocal scale factor.

For example, if a 6 cm side corresponds to a 15 cm side, the scale factor from the first shape to the second is

Every length on the second shape is therefore 2.5 times the corresponding length on the first. A corresponding 9 cm side becomes $9\times2.5=22.5$ cm. In the reverse direction, divide by 2.5, or multiply by $1/2.5$.

The same idea can be written using equal ratios. If triangles $ABC$ and $PQR$ are similar and $AB$, $BC$ and $AC$ correspond to $PQ$, $QR$ and $PR$, then

Each fraction must compare matching sides in the same order.

Why area uses the square of the scale factor

An area has two dimensions. If every length is multiplied by $k$, both dimensions contributing to the area are multiplied by $k$. The area is therefore multiplied by

For similar shapes:

This applies to the whole area of a two-dimensional shape and to corresponding surface areas of similar solids. If the length scale factor is 3, the area scale factor is $3^2=9$, not 3. A rectangle whose length and width are both tripled covers nine times the original area.

Why volume uses the cube of the scale factor

A volume has three dimensions. For similar solids, length, width and height are each multiplied by $k$, so the volume is multiplied by

Therefore,

If every length is doubled, the volume becomes $2^3=8$ times as large. If every length is halved, the volume becomes $\left(\frac12\right)^3=\frac18$ of the original.

Moving between ratios

If corresponding lengths are in the ratio $a:b$, then corresponding areas and volumes are in the ratios

For example, a length ratio of $2:5$ produces an area ratio of $2^2:5^2=4:25$ and a volume ratio of $2^3:5^3=8:125$. Keep the order unchanged throughout: these are all first shape : second shape.

Working backwards from area or volume

Sometimes the area or volume scale factor is given instead of the length scale factor. Reverse the power operation:

and

For instance, if the areas of two similar shapes are 45 cm² and 180 cm², the area scale factor from the smaller to the larger is $180/45=4$. The length scale factor is therefore $\sqrt4=2$. A 6 cm length on the smaller shape corresponds to a 12 cm length on the larger.

If the volume ratio of two similar solids is $27:64$, the corresponding length ratio is

Once the length factor is known, it can also be squared to compare corresponding surface areas. This allows information about volume, surface area and length to be linked without finding every dimension separately.

A reliable method

For any similarity calculation:

1. Decide the direction of comparison, such as small to large.
2. Match corresponding measurements.
3. Form the appropriate factor: length, area or volume.
4. Convert between factors using a square, cube, square root or cube root if necessary.
5. Multiply the known measurement by the factor.
6. Check that the size and units of the answer make sense.

Lengths use ordinary units such as cm, areas use square units such as cm², and volumes use cubic units such as cm³.