Similar Figures and Scale Factors

The key idea

If two figures are similar with scale factor

k

, lengths scale by

k

, areas by

k^{2}

, and volumes by

k^{3}

. Find

k

from whichever measurement you are given, then convert.

The trap

Scaling area or volume by $k$ itself.

Can Desmos do it?

Desmos can't see the figures. Pick the right power of $k$ .

College Board skill: Area and volume · Geometry and Trigonometry

Try one at each level

One question from each level. The drill has 12 at every level, and moves you up after four right in a row.

Level 1 · Warm-up

Triangle

A B C

is similar to triangle

D E F

, with

A B

corresponding to

D E

. If

A B = 4

D E = 6

, and the area of triangle

A B C

is 32, what is the area of triangle

D E F

Show the answer

Answer: 72The scale factor from

A B C

D E F

\frac{6}{4}

, so areas scale by

(\frac{6}{4})^{2} = \frac{9}{4}

. The area is

32 \times \frac{9}{4} = 72

Level 2 · Test level

Two cylinders are similar. Their surface areas are

25 π

and

16 π

square units, and the volume of the first cylinder is

625 π

cubic units. The volume of the second cylinder is

k π

cubic units. What is the value of

k

Show the answer

Answer: 320The ratio of surface areas is

\frac{16}{25} = \frac{16}{25} = (\frac{4}{5})^{2}

, so the length scale factor is

\frac{4}{5}

and volumes scale by

(\frac{4}{5})^{3} = \frac{64}{125}

. So

k = 625 \times \frac{64}{125} = 320

Level 3 · Beyond the test

A right circular cone has a height of 6. A plane parallel to the base cuts the cone 3 units above the base. The piece between the plane and the base is what fraction of the cone's volume?

Show the answer

Answer: 7/8The small cone above the cut has height

6 - 3 = 3

, so its scale factor is

\frac{1}{2}

and its volume is

(\frac{1}{2})^{3} = \frac{1}{8}

of the original. The bottom piece is the rest:

1 - \frac{1}{8} = \frac{7}{8}

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