Math Core

Similar figures have the same shape but possibly different sizes. Similarity is the tool for any problem with parallel lines inside a triangle, shadows, maps and scale models, or two figures where one is a blown-up copy of the other. The whole topic rests on one number: the scale factor.

Similar triangles

Two triangles are similar when their angles match. Then every pair of corresponding sides has the same ratio, called the scale factor.

Since the angles of a triangle add to $180^\circ$ , you only need two pairs of equal angles to know the triangles are similar (the AA rule). Look for these common sources of equal angles:

A shared angle, like the angle at the top of a triangle cut by a line.
Parallel lines, which make corresponding and alternate interior angles equal.
Vertical angles, where two segments cross.
Right angles, especially an altitude drawn to the hypotenuse of a right triangle, which creates three similar right triangles.

When you write $\triangle ABC \sim \triangle DEF$ , the order matters: $A$ matches $D$ , $B$ matches $E$ , and $C$ matches $F$ . Then $\dfrac{DE}{AB} = \dfrac{EF}{BC} = \dfrac{DF}{AC}$ .

Scale factor rules

Lengths, areas and volumes

If two similar figures have scale factor $k$ (every length in the second is $k$ times the matching length in the first), then:

lengths (sides, heights, perimeters) are in the ratio $k$ ,
areas are in the ratio $k^2$ ,
volumes are in the ratio $k^3$ .

This works for any similar figures, not just triangles: circles, polygons, cones, even statues. Going backward, if the areas are in the ratio $9 : 25$ , the lengths are in the ratio $3 : 5$ .

Worked example: A shadow

A $6$ -foot person casts a $4$ -foot shadow. At the same time, a tree casts a $30$ -foot shadow. How tall is the tree?

The sun's rays hit both at the same angle, and both stand straight up, so the two triangles (object, shadow, ray) are similar. The ratio of height to shadow is $\dfrac{6}{4} = \dfrac{3}{2}$ , so the tree is $\dfrac{3}{2} \cdot 30 = 45$ feet tall.

Worked example: A segment parallel to a side

In triangle $ABC$ , $D$ is on $AB$ and $E$ is on $AC$ with $\overline{DE} \parallel \overline{BC}$ . If $AD = 4$ , $DB = 6$ and $DE = 5$ , find $BC$ . What fraction of triangle $ABC$ 's area lies below $DE$ ?

Triangle ABC with A at the top. Segment DE is parallel to base BC.

The parallel lines make $\angle ADE = \angle ABC$ , and angle $A$ is shared, so $\triangle ADE \sim \triangle ABC$ . The scale factor is $\dfrac{AB}{AD} = \dfrac{10}{4} = \dfrac{5}{2}$ . (Use the whole side $AB = 4 + 6$ , not $DB$ .) So $BC = \dfrac{5}{2} \cdot 5 = 12.5$ .

The areas are in the ratio $\left(\dfrac{2}{5}\right)^2 = \dfrac{4}{25}$ . The small triangle is $\dfrac{4}{25}$ of the big one, so the piece below $DE$ is $\dfrac{21}{25}$ of triangle $ABC$ .

Worked example: Diagonals of a trapezoid

Trapezoid $ABCD$ has bases $AB = 12$ and $CD = 4$ and height $8$ . Its diagonals meet at $P$ . Find the areas of the four triangles the diagonals create.

Trapezoid with bases 12 (bottom) and 4 (top). The diagonals cross at P.

Triangles $PAB$ and $PCD$ are similar (vertical angles at $P$ , and alternate interior angles from the parallel bases), with scale factor $\dfrac{12}{4} = 3$ . So their heights are in the ratio $3 : 1$ and add to $8$ : the heights are $6$ and $2$ .

$\triangle PAB$ : $\dfrac{1}{2} \cdot 12 \cdot 6 = 36$ .
$\triangle PCD$ : $\dfrac{1}{2} \cdot 4 \cdot 2 = 4$ .
$\triangle ABC$ has area $\dfrac{1}{2} \cdot 12 \cdot 8 = 48$ , so $\triangle PBC = 48 - 36 = 12$ . By the same reasoning, $\triangle PDA = 12$ .

Check: $36 + 4 + 12 + 12 = 64 = \dfrac{1}{2}(12 + 4) \cdot 8$ . Notice $12 \cdot 12 = 36 \cdot 4$ : the two side triangles always have equal areas, and each is the geometric mean of the other two.

Worked example: Similar solids

Two similar cones have heights $6$ and $9$ . The smaller has volume $16\pi$ . Find the volume of the larger.

The scale factor is $\dfrac{9}{6} = \dfrac{3}{2}$ , so volumes are in the ratio $\left(\dfrac{3}{2}\right)^3 = \dfrac{27}{8}$ . The larger volume is $\dfrac{27}{8} \cdot 16\pi = 54\pi$ .

Common mistake

When a line parallel to the base cuts off a small triangle, compare the small triangle's side to the whole side of the big triangle, not to the leftover piece. With $AD = 4$ and $DB = 6$ , the ratio is $4 : 10$ , not $4 : 6$ .

Practice

Practice 1

A triangle has sides $4$ , $6$ and $8$ . A similar triangle has shortest side $10$ . What is its longest side?