Math Core

Maps, floor plans and model cars all shrink something big down to a size you can hold. A scale drawing does it so every length shrinks by the same amount, which means you can measure the drawing and figure out the real size.

What a scale tells you

A scale drawing always comes with a scale that links drawing lengths to actual lengths. You might see it written in a few ways:

$1 \text{ cm} : 5 \text{ m}$ (1 centimeter on the drawing stands for 5 meters in real life)
$1 \text{ inch} = 20 \text{ miles}$
$1 : 50$ (1 unit on the drawing stands for 50 of the same unit in real life)

Because every length is shrunk the same way, drawing length and actual length are in a proportional relationship. The scale is the unit rate.

Definition

Scale

The scale of a drawing compares a length on the drawing to the matching actual length. When both lengths use the same unit, the scale can be written as a ratio like $1 : 50$ , and $50$ is the scale factor from the drawing to the real object.

A scale drawing of a garden, 5 cm by 3 cm. The scale is 1 cm : 4 m.

Finding actual lengths

To go from the drawing to real life, multiply by the number the scale gives for $1$ unit. You can also set up a proportion, which is handy when the scale isn't "1 to something."

Worked example: Reading a map

On a map, $1$ cm represents $15$ km. Two towns are $6.4$ cm apart on the map. How far apart are they really?

Each centimeter stands for $15$ km, so

6.4 \times 15 = 96 \text{ km}.

Worked example: A scale that isn't 1 to something

On a blueprint, $2$ inches represent $5$ feet. A wall is $9$ inches long on the blueprint. How long is the real wall?

Set up a proportion with drawing lengths on top:

\frac{2 \text{ in}}{5 \text{ ft}} = \frac{9 \text{ in}}{x \text{ ft}}

Since $9 \div 2 = 4.5$ , the drawing length was multiplied by $4.5$ , so the actual length is $5 \times 4.5 = 22.5$ feet.

Area in a scale drawing

Look at the garden above with scale $1$ cm : $4$ m. The real garden is $5 \times 4 = 20$ m long and $3 \times 4 = 12$ m wide, so its area is $20 \times 12 = 240$ square meters.

The drawing's area is only $5 \times 3 = 15$ square centimeters. Each square centimeter on the drawing stands for a $4$ m by $4$ m square, which is $16$ square meters. And $15 \times 16 = 240$ . It checks out.

Common mistake

Don't use the scale directly on an area. If $1$ cm stands for $4$ m, then $1$ square centimeter stands for $4 \times 4 = 16$ square meters, not $4$ . The safest method: find the actual lengths first, then multiply them.

Redrawing at a different scale

Sometimes you need to redraw a picture at a new scale, maybe to fit a bigger poster. Go through the actual size: first find the real length, then use the new scale to find the new drawing length.

Worked example: Changing the scale

A drawing of a soccer field uses the scale $1$ cm : $10$ m, and the field is $10.5$ cm long. You want to redraw it with the scale $1$ cm : $7$ m. How long will the field be in the new drawing?

Actual length: $10.5 \times 10 = 105$ m.

New drawing: every $7$ m becomes $1$ cm, so $105 \div 7 = 15$ cm.

The new drawing is bigger, which makes sense: each centimeter now stands for less distance.

Tip

Check that your answer makes sense. Real objects are almost always larger than their drawings, so if you multiply and get something smaller, you probably divided when you should have multiplied.

Practice

Practice 1

On a map, $1$ cm represents $20$ km. Two lakes are $4.5$ cm apart on the map. How many kilometers apart are the lakes?