Scale Drawings: Find Real and Drawing Lengths | Grade 8 Math | LevelKite

A scale drawing shows a real object, such as a garden, a room or a town, with the same shape but a different size, usually smaller. Every length is shrunk (or enlarged) in the same proportion.

Each scale drawing has a scale, a key that says what a length on the drawing represents. For example, the scale " $1$ cm represents $4$ m" means that each centimeter on the drawing stands for $4$ meters of the real object. A length on the drawing and the real length it stands for are corresponding lengths.

Every drawing length in this lesson is stated for you. Never measure a drawing on your screen. A screen can show a drawing larger or smaller than its true size, but the stated lengths stay the same. Below is a plan of a garden drawn with the scale " $1$ cm represents $4$ m".

Scale: 1 centimeter on the drawing represents 4 meters. Segment A B is 8 centimeters on the drawing and represents 32 meters. Segment B C is 6 centimeters on the drawing and represents 24 meters. Segment A C is 10 centimeters on the drawing and represents 40 meters. Drawing lengths are given, not measured on the screen.

Garden

ABCD

with path

AC

. Scale:

1

cm represents

4

m.

The plan shows a rectangular garden $ABCD$ with a straight path from corner $A$ to corner $C$ . Side $AB$ is $8$ cm on the plan. Each of those $8$ centimeters stands for $4$ m, so the real side is $8 \times 4 = 32$ m long. In the same way:

Side $BC$ is $6$ cm on the plan, so the real side is $6 \times 4 = 24$ m long.
Path $AC$ is $10$ cm on the plan, so the real path is $10 \times 4 = 40$ m long.

The path is slanted, but the scale works for it too. Every length on a scale drawing follows the scale, in any direction. Divide each real length by its plan length, which is never zero, and you get the same result every time:

\frac{32\text{ m}}{8\text{ cm}} = \frac{24\text{ m}}{6\text{ cm}} = \frac{40\text{ m}}{10\text{ cm}} = \frac{4\text{ m}}{1\text{ cm}}

Key concept

Real lengths are proportional to drawing lengths

Suppose a scale says that $a$ units on the drawing represent $b$ real units, where $a$ and $b$ are positive. Then every pair of corresponding lengths has the same ratio as the scale: the real length divided by the drawing length is always $\frac{b}{a}$ . This number is the real length for $1$ drawing unit, a unit rate.

Drawing to real: multiply the drawing length by $\frac{b}{a}$ .
Real to drawing: divide the real length by $\frac{b}{a}$ . The rate is positive, so it is never zero and the division is defined.

Keep each length in the unit the scale uses for it: drawing units for drawing lengths, real units for real lengths.

\frac{\text{real length}}{\text{drawing length}} = \frac{b}{a} \qquad (a > 0,\ b > 0)

Worked example

From a map length to a real length

On a road map, $1$ centimeter represents $3$ kilometers. A straight stretch of highway is $4.5$ centimeters long on the map. How long is the real stretch of highway?

Write the scale as a rate. Each centimeter on the map stands for 3 kilometers, so the rate is 3 kilometers per centimeter.
$\frac{3\text{ km}}{1\text{ cm}}$
Multiply the map length by the rate. The 4 whole centimeters stand for 4 times 3 km, and the half centimeter stands for half of 3 km, so 4.5 cm stands for 4.5 times 3 km. The centimeters cancel, so the result is in kilometers.
$4.5\text{ cm} \times \frac{3\text{ km}}{1\text{ cm}} = 13.5\text{ km}$
Check that the size makes sense. The map length is between 4 cm and 5 cm, so the real length must be between 4 times 3 km and 5 times 3 km.
$12\text{ km} < 13.5\text{ km} < 15\text{ km}$

Answer

The real stretch of highway is $13.5$ kilometers long.

Worked example

From a real length to a plan length

Maya is drawing a floor plan of her school library. The library is a rectangle $54$ feet long and $36$ feet wide. On her plan, $1$ inch represents $6$ feet. How long and how wide should she draw the library on her plan?

Decide which way to use the scale. Each inch on the plan stands for 6 feet, so the number of inches on the plan is the number of 6-foot lengths in the real length. That is a division by 6, which is not zero.
$\text{plan length in inches} = \text{real length in feet} \div 6$
Find the length on the plan. Dividing by the rate 6 feet per inch is the same as multiplying by its reciprocal, 1 inch per 6 feet. The feet cancel, so the result is in inches.
$54\text{ ft} \times \frac{1\text{ in}}{6\text{ ft}} = \frac{54}{6}\text{ in} = 9\text{ in}$
Find the width on the plan the same way.
$36\text{ ft} \times \frac{1\text{ in}}{6\text{ ft}} = \frac{36}{6}\text{ in} = 6\text{ in}$
Check by turning each plan length back into a real length with the scale. 9 inches stands for 9 times 6 feet, and 6 inches stands for 6 times 6 feet. Both results match the library.
$9 \times 6\text{ ft} = 54\text{ ft}, \qquad 6 \times 6\text{ ft} = 36\text{ ft}$

Answer

Maya should draw the library $9$ inches long and $6$ inches wide.

Scale: 1 inch on the drawing represents 6 feet. Segment A B is 9 inches on the drawing and represents 54 feet. Segment B C is 6 inches on the drawing and represents 36 feet. Drawing lengths are given, not measured on the screen.

Maya's plan: length

AB

is

9

in and width

BC

is

6

in. Scale:

1

in represents

6

ft.

Some scales do not start with $1$ , such as " $2$ cm represents $500$ m". Divide to find the real length for $1$ cm: $500 \div 2 = 250$ , so each centimeter stands for $250$ m. The divisor $2$ is a drawing length, so it is not zero. You can also skip the rate and write a proportion, with drawing lengths in one position and real lengths in the other.

Either way, use the units of the scale. If a length is given in another unit, convert it first. For example, when the scale uses meters, change kilometers to meters using $1$ km $= 1000$ m.

Worked example

A scale that does not start with 1

A town map uses the scale " $2$ cm represents $500$ m". A new straight bike trail will be $3.2$ kilometers long. How long should the trail be drawn on the map?

Make the units match the scale. The scale gives real lengths in meters, and 1 kilometer is 1000 meters, so change the trail's length to meters.
$3.2\text{ km} = 3.2 \times 1000\text{ m} = 3200\text{ m}$
Define the unknown. Let d be the trail's length on the map, in centimeters. A length is positive, so d is greater than 0.
$d > 0$
Write a proportion. Map lengths in centimeters go on top and real lengths in meters go on the bottom, in both ratios. The denominators 500 and 3200 are not zero, so both ratios are defined.
$\frac{2\text{ cm}}{500\text{ m}} = \frac{d\text{ cm}}{3200\text{ m}}$
Write the cross products. This is the same as multiplying both sides by 500 times 3200, which is not zero, so the solution does not change.
$500 \cdot d = 2 \cdot 3200 = 6400$
Divide both sides by 500, which is not zero, to get d by itself.
$d = \frac{6400}{500} = 12.8$
Check with the rate for 1 cm, which is 500 divided by 2, or 250 meters per centimeter. Then 12.8 cm stands for 12.8 times 250 meters, which is 3200 meters, or 3.2 kilometers.
$12.8 \times 250\text{ m} = 3200\text{ m} = 3.2\text{ km}$

Answer

The trail should be drawn $12.8$ centimeters long on the map.

Takeaway. A scale drawing has the same shape as the real object, and its scale gives every pair of corresponding lengths the same ratio. So real lengths are proportional to drawing lengths. To find a real length, multiply the drawing length by the real length for $1$ drawing unit. To find a drawing length, divide the real length by that same rate, which is never zero, or solve a proportion with matching units in matching positions. Convert units to match the scale first, use only stated lengths, and check each answer by turning it back with the scale.