Chapter 3 · Ratios and proportional relationships

Missing Lengths and Word Problems in Scale Drawings

Use one scale to find a missing side in a scale drawing of a polygon, then solve word problems about maps and drawings by keeping corresponding lengths and units matched.

20 min read

You already know how to use a scale to turn a length on a drawing into an actual length. In this lesson you will find a length that is missing, either on the actual figure or on the drawing. Then you will use the same thinking in word problems.

Three terms will help:

A scale drawing is a drawing of an actual figure in which one scale sets every length.
A scale is a ratio (a comparison of two quantities). It compares a length on the drawing with the actual length it stands for, with the drawing length written first. The scale $1$ cm : $2$ cm means each $1$ cm on the drawing stands for $2$ cm on the actual figure.
Corresponding sides are a side of the drawing and the side of the actual figure that it stands for. They sit in matching positions.

A scale always compares corresponding sides, never two sides in different positions.

The figure below shows two polygons (closed shapes with straight sides) on a grid. Each grid step on the horizontal axis ( $x$ ) and on the vertical axis ( $y$ ) is $1$ cm. The small polygon on the left is a scale drawing. The large polygon on the right is the actual figure. The scale is $1$ cm : $2$ cm, drawing to actual.

You can measure a side that lies along a grid line by subtracting coordinates, larger minus smaller, so the length is positive. For a horizontal side, subtract the $x$ -coordinates of its endpoints. For a vertical side, subtract the $y$ -coordinates. Each grid step is $1$ cm, so the difference is the length in centimeters. The bottom side of the drawing goes from $(1, 1)$ to $(7, 1)$ , so it is $7 - 1 = 6$ cm long.

The slanted sides do not lie along grid lines, so you cannot measure them this way. Here is one more fact: the slanted side of the drawing is $5$ cm long. The slanted side of the actual figure is the missing length.

The small polygon is the scale drawing and the large polygon is the actual figure. Each grid step is

1

cm, and the scale is

1

cm :

2

cm.

Corresponding sides, measured from the grid

Side	Drawing (small polygon)	Actual figure (large polygon)	Actual divided by drawing
Bottom	$(1, 1)$ to $(7, 1)$ : $7 - 1 = 6$ cm	$(9, 1)$ to $(21, 1)$ : $21 - 9 = 12$ cm	$12 \div 6 = 2$
Top	$(1, 4)$ to $(3, 4)$ : $3 - 1 = 2$ cm	$(9, 7)$ to $(13, 7)$ : $13 - 9 = 4$ cm	$4 \div 2 = 2$
Left	$(1, 1)$ to $(1, 4)$ : $4 - 1 = 3$ cm	$(9, 1)$ to $(9, 7)$ : $7 - 1 = 6$ cm	$6 \div 3 = 2$
Slanted	$(7, 1)$ to $(3, 4)$ : given as $5$ cm	$(21, 1)$ to $(13, 7)$ : missing	not known yet

Look across the three rows you can measure. In each one, the actual side is $2$ times as long as its corresponding drawing side, just as the scale $1$ cm : $2$ cm says. The number $2$ is the scale factor from the drawing to the actual figure: the number you multiply a drawing length by to get the actual length, when both lengths use the same unit.

The scale factor also takes you back. The actual top side is $4$ cm long, and $4 \div 2 = 2$ cm is the length of the drawing's top side.

In a scale drawing, one pair of corresponding sides that you can measure is enough to find a scale that nobody gave you: $12 \div 6 = 2$ .

Key concept

One scale factor links every pair of corresponding lengths

A scale compares a drawing length with the actual length of its corresponding side. When both lengths are positive and written in the same unit, the scale factor from the drawing to the actual figure is the actual length divided by the drawing length. The drawing length is the divisor (the number you divide by) and is never zero, so the scale factor tells you the actual length for each $1$ unit of drawing length.

Every pair of corresponding sides has the same scale factor:

Drawing to actual: multiply the drawing length by the scale factor.
Actual to drawing: divide the actual length by the scale factor, which is positive and so not zero.

This rule is for lengths. Areas multiply by the square of the scale factor instead.

\text{actual length} = \text{scale factor} \times \text{drawing length}

Worked example

The missing slanted side

In the figure of the two polygons above, the small polygon is a scale drawing of the large polygon, which is the actual figure. Each grid step is $1$ cm. The slanted side of the drawing is $5$ cm long. Find the length of the slanted side of the actual figure.

Choose a pair of corresponding sides you can measure on the grid: the bottom sides. They are horizontal, so subtract the x-coordinates of their endpoints. The drawing's bottom side goes from (1, 1) to (7, 1). The actual figure's bottom side goes from (9, 1) to (21, 1). Each grid step is 1 cm, so each difference is a length in centimeters.
$7 - 1 = 6 \text{ cm} \qquad 21 - 9 = 12 \text{ cm}$
Find the scale factor from the drawing to the actual figure. Both lengths are positive and in centimeters, so divide the actual length by the drawing length. The divisor 6 is not zero. The result agrees with the scale 1 cm : 2 cm.
$\frac{12 \text{ cm}}{6 \text{ cm}} = 2$
Let s be the length of the slanted side of the actual figure, in centimeters. The two slanted sides are corresponding sides, so the same scale factor links them. Each 1 cm of the drawing stands for 2 cm, so the 5 cm side stands for 5 groups of 2 cm. Multiply 5 by 2.
$s = 2 \times 5 = 10$
Check that the answer is reasonable. The scale factor is greater than 1 and both lengths are in centimeters, so the actual side must be longer than the 5 cm drawing side. A length of 10 cm is longer, so the answer fits.

Answer

The slanted side of the actual figure is $10$ cm long.

Many word problems give the scale with units, such as $2$ cm : $5$ km, and the two units can be different. Every length is a positive number with a unit. Keep the unit beside every number so you can see what each number measures.

A proportion is an equation that says two ratios are equal. Every pair of corresponding lengths has the same ratio as the scale, so you can set the two ratios equal. Put the drawing lengths in the same position in both ratios, such as both on top, and the actual lengths in the other position. Use one unit for both drawing lengths and one unit for both actual lengths. Then the units match across the equals sign, and solving the proportion gives the missing length, whether it is on the drawing or on the actual object.

Worked example

A bike path on a map

A map of a state park is a scale drawing with the scale $2$ cm : $5$ km. This means $2$ cm on the map stands for $5$ km of actual distance. A bike path is actually $15$ km long. How long is the bike path on the map?

Let d be the length of the bike path on the map, in centimeters. The path and the scale are on the same map, so both ratios of map length to actual length are equal. Write each ratio with the map length on top and the actual length on the bottom: d goes with 15 km, and 2 cm goes with 5 km.
$\frac{d \text{ cm}}{15 \text{ km}} = \frac{2 \text{ cm}}{5 \text{ km}}$
Both sides are centimeters over kilometers, so the units match and you can work with the numbers. Multiply both sides by 15. Doing the same thing to both sides keeps them equal, and because 15 is not zero the new equation has the same solution. On the left, dividing by 15 and then multiplying by 15 leaves d alone.
$\frac{d}{15} \times 15 = \frac{2}{5} \times 15$
Do the arithmetic on the right: 2 times 15 is 30, and 30 divided by 5 is 6. The result is positive, as a length must be.
$d = \frac{2 \times 15}{5} = \frac{30}{5} = 6$
Check with the meaning of the scale. The actual 15 km is 3 groups of 5 km, because 15 divided by 5 is 3. Each group takes 2 cm on the map, so the path takes 3 groups of 2 cm. Both methods give 6 cm.
$3 \times 2 \text{ cm} = 6 \text{ cm}$

Answer

The bike path is $6$ cm long on the map.

Sometimes two corresponding lengths come in different units, like $12$ cm on a drawing and $6$ m on the actual object. You can compare them as a rate (a comparison of two quantities with different units), such as centimeters for each $1$ meter.

A scale factor is different. It is a plain number with no unit, so it only makes sense when both lengths use the same unit. If you divided $6$ by $12$ without matching the units, you would get $0.5$ , which would wrongly say the actual object is half as long as its drawing. So rewrite one length first, using a conversion fact such as $1$ m $= 100$ cm. Each problem in this lesson gives you the fact you need.

Worked example

A garden drawing with mixed units

Ana makes a scale drawing of a triangular garden. One side of the garden is $6$ m long, and she draws it $12$ cm long. Another side of the garden is $10$ m long. How long should she draw that side? What is the scale factor from the drawing to the actual garden? Use $1$ m $= 100$ cm.

Start with the pair of corresponding sides you know both ways: 12 cm on the drawing stands for 6 m in the garden. Find the drawing length for each 1 m by dividing both numbers by 6. The 6 m is the amount that becomes 1, so 6 is the divisor, and it is not zero. This gives the scale 2 cm : 1 m.
$\frac{12 \text{ cm}}{6 \text{ m}} = \frac{2 \text{ cm}}{1 \text{ m}}$
Let d be the drawing length of the 10 m side, in centimeters. Every pair of corresponding sides uses the same scale, so each of the 10 meters takes 2 cm on the drawing. Multiply 10 by 2.
$d = 10 \times 2 = 20$
A scale factor compares two lengths in the same unit, so write the actual 6 m side in centimeters. Each 1 m is 100 cm, so multiply the number of meters by 100.
$6 \text{ m} = 6 \times 100 \text{ cm} = 600 \text{ cm}$
Divide the actual length by the drawing length of the same side. Both lengths are positive and in centimeters, and the divisor 12 is not zero.
$\frac{600 \text{ cm}}{12 \text{ cm}} = 50$
Check with the side you found. Multiply its drawing length, 20 cm, by the scale factor 50. Every 100 cm make 1 m, so divide the number of centimeters by 100 to write the result in meters. You get the actual 10 m, so the two answers fit together.
$50 \times 20 \text{ cm} = 1{,}000 \text{ cm} = 10 \text{ m}$

Answer

Ana should draw the $10$ m side $20$ cm long. The scale factor from the drawing to the actual garden is $50$ , so each actual length is $50$ times its drawing length when both are in the same unit.

Here is the idea to keep. In a scale drawing, one scale links every pair of corresponding lengths. Use the scale you are given, or find it from a pair of corresponding sides you know, with the units kept and matched. Then multiply a drawing length by the scale factor to get the actual length, divide an actual length by the scale factor to get the drawing length, or solve a proportion with drawing lengths and actual lengths in matching positions. This rule is for lengths.

Next, you will compute actual areas from a scale drawing, where the square of the scale factor matters, and you will reproduce a scale drawing at a different scale.

Up next

Actual Areas and Redrawing at a New Scale

20 min read

Continue