Scale, Scale Factor and Actual Lengths | Grade 7 Math | LevelKite

A scale drawing is a drawing of a real object in which every length of the object has been multiplied by the same positive number. Maps, floor plans and blueprints are scale drawings. Because every length changes by the same multiplier, the drawing has exactly the same shape as the real object. Only the size is different.

Each side of the drawing matches one side of the real object. Two sides that match like this are called corresponding sides. In this lesson, you will see how the lengths of corresponding sides are linked, and you will use that link to find actual lengths from a drawing.

The scale of a drawing is a ratio that compares a length on the drawing to the actual length it stands for. The drawing length is written first. For example, the scale $1\text{ cm} : 2\text{ m}$ means that every $1$ cm on the drawing stands for $2$ m on the real object.

The same scale holds for every pair of corresponding sides, so drawing length and actual length are in a proportional relationship. A side that is $3$ cm on the drawing is $3$ groups of $1$ cm, so it stands for $3 \times 2 = 6$ m.

The table lists three pairs of corresponding lengths for this scale. It also writes each actual length in centimeters, using $1\text{ m} = 100\text{ cm}$ , so that both lengths have the same unit. Then it divides the actual length by the drawing length.

Three pairs of corresponding lengths for the scale 1 cm : 2 m

Drawing length	Actual length	Actual length in cm	Actual ÷ drawing (both in cm)
$1$ cm	$1 \times 2 = 2$ m	$2 \times 100 = 200$ cm	$200 \div 1 = 200$
$3$ cm	$3 \times 2 = 6$ m	$6 \times 100 = 600$ cm	$600 \div 3 = 200$
$5$ cm	$5 \times 2 = 10$ m	$10 \times 100 = 1000$ cm	$1000 \div 5 = 200$

In every row of the table, once both lengths are in centimeters, the actual length is $200$ times the drawing length. That constant multiplier has a name.

The scale factor from the drawing to the actual object is the number you multiply a drawing length by to get the corresponding actual length, when both lengths are in the same unit. To find it, write both parts of the scale in the same unit and divide the actual part by the drawing part. The scale $1\text{ cm} : 2\text{ m}$ becomes $1\text{ cm} : 200\text{ cm}$ , so the scale factor is $200\text{ cm} \div 1\text{ cm} = 200$ .

A scale factor has no unit, because dividing centimeters by centimeters cancels the unit. It is positive, because lengths are positive. This one is greater than $1$ because the real object is larger than the drawing. Going the other way, from the actual object to the drawing, you multiply by $\frac{1}{200}$ instead, which undoes multiplying by $200$ .

Worked example

From a scale to a scale factor

A scale drawing of a school building uses the scale $2\text{ cm} : 6\text{ m}$ . Find the scale factor from the drawing to the actual building.

Read the scale in order: the drawing length comes first and the actual length second. A side that is 2 cm long on the drawing corresponds to a side that is 6 m long on the real building.
$2\text{ cm} : 6\text{ m}$
Write both lengths in the same unit so that you can compare them. There are 100 cm in each meter, so multiply 6 by 100. This changes how the length is written, not how long it is.
$6\text{ m} = 6 \times 100\text{ cm} = 600\text{ cm}$
Divide the actual length by the drawing length. Both are in centimeters, so the unit cancels and a plain number is left. You can divide because the drawing length is not zero.
$600\text{ cm} \div 2\text{ cm} = 300$
Check the result against the scale. Multiplying the 2 cm drawing length by 300 should give the 6 m actual length. It gives 600 cm, and 600 divided by 100 is 6, so it is 6 m.
$2\text{ cm} \times 300 = 600\text{ cm} = 6\text{ m}$

Answer

The scale factor from the drawing to the actual building is $300$ . Every actual length is $300$ times its corresponding drawing length when both are in the same unit.

Key concept

Scale, scale factor and actual length

A scale compares a drawing length to the actual length it stands for. It holds for every pair of corresponding sides.

Write both parts of the scale in the same unit and divide the actual part by the drawing part. The result is the scale factor from the drawing to the actual object.

For corresponding sides measured in the same unit, the actual length is the scale factor times the drawing length.

This rule is for lengths only. An area is measured with two lengths, such as a length and a width, and each one is multiplied by the scale factor. So an actual area is the drawing's area times the square of the scale factor, when both areas use the same square unit.

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

Now read a scale drawing made on a coordinate grid. The rectangle in the figure below is a scale drawing of a playground. The horizontal axis is the $x$ -axis and the vertical axis is the $y$ -axis. One grid unit is the distance between two neighboring grid lines. The drawing is on centimeter grid paper, so each grid unit is $1$ cm long.

Every side of this rectangle lies on a grid line, so you can find its drawing length from the coordinates of its two ends:

A horizontal side has ends with the same $y$ -coordinate. Its length in grid units is the difference of the $x$ -coordinates.
A vertical side has ends with the same $x$ -coordinate. Its length in grid units is the difference of the $y$ -coordinates.

Subtract the smaller coordinate from the larger one, so that the length is positive.

Each grid unit is

1

cm on the drawing. The rectangle has corners

(1, 1)

,

(7, 1)

,

(7, 5)

and

(1, 5)

, and the scale is

1\text{ cm} : 4\text{ m}

.

Worked example

Actual lengths of a rectangle

The rectangle in the figure above is a scale drawing of a playground. Each grid unit is $1$ cm on the drawing, and the scale is $1\text{ cm} : 4\text{ m}$ . The corners are at $(1, 1)$ , $(7, 1)$ , $(7, 5)$ and $(1, 5)$ . Find the actual length and width of the playground.

Find the drawing length of the bottom side. Its ends are (1, 1) and (7, 1). They have the same y-coordinate, so the side is horizontal. Subtract the x-coordinates. Each grid unit is 1 cm, so 6 grid units make 6 cm.
$7 - 1 = 6\text{ grid units} = 6\text{ cm}$
Find the drawing length of the right side. Its ends are (7, 1) and (7, 5). They have the same x-coordinate, so the side is vertical. Subtract the y-coordinates. Opposite sides of a rectangle are equal, so the top is also 6 cm and the left side is also 4 cm.
$5 - 1 = 4\text{ grid units} = 4\text{ cm}$
Find the scale factor from the drawing to the playground. Write 4 m in centimeters, using 100 cm in each meter, so that both parts of the scale have the same unit. Then divide the actual part by the drawing part.
$4\text{ m} = 4 \times 100\text{ cm} = 400\text{ cm}, \quad 400\text{ cm} \div 1\text{ cm} = 400$
Multiply each drawing length by the scale factor. This is valid because each drawing side and the real side it stands for are corresponding sides, and both lengths are in centimeters.
$6 \times 400 = 2400\text{ cm}, \quad 4 \times 400 = 1600\text{ cm}$
Write the lengths in meters, a better unit for a playground. There are 100 cm in each meter, so divide each number of centimeters by 100.
$2400 \div 100 = 24\text{ m}, \quad 1600 \div 100 = 16\text{ m}$
Check with the scale itself. Each 1 cm on the drawing stands for 4 m, so 6 cm stands for 6 groups of 4 m and 4 cm stands for 4 groups of 4 m. Both results agree with the step before.
$6 \times 4 = 24\text{ m}, \quad 4 \times 4 = 16\text{ m}$

Answer

The actual playground is $24$ m long and $16$ m wide.

Each grid unit is

1

cm on the drawing. The triangle has corners

(1, 1)

,

(9, 1)

and

(9, 7)

. Its slanted side is given as

10

cm, and the scale is

2\text{ cm} : 5\text{ m}

.

Worked example

A triangle with a slanted side

The triangle in the figure above is a scale drawing of a park. The horizontal axis is the $x$ -axis and the vertical axis is the $y$ -axis. Each grid unit is $1$ cm on the drawing, and the scale is $2\text{ cm} : 5\text{ m}$ . The corners are at $(1, 1)$ , $(9, 1)$ and $(9, 7)$ . The slanted side, from $(1, 1)$ to $(9, 7)$ , is given as $10$ cm long on the drawing. Find the actual length of each side.

Find the drawing lengths. The bottom side, from (1, 1) to (9, 1), is horizontal, so subtract the x-coordinates. The right side, from (9, 1) to (9, 7), is vertical, so subtract the y-coordinates. Each grid unit is 1 cm, so each difference is a length in centimeters. The slanted side does not lie on a grid line, so one subtraction cannot give its length. Use the given 10 cm.
$9 - 1 = 8\text{ cm}, \quad 7 - 1 = 6\text{ cm}, \quad \text{slanted side} = 10\text{ cm}$
Let b be the actual length of the bottom side, in meters. The scale holds for every pair of corresponding sides, so the ratio of drawing length to actual length equals the scale. Write a proportion. On both sides, put drawing centimeters on top and actual meters on the bottom, so the units match.
$\frac{2\text{ cm}}{5\text{ m}} = \frac{8\text{ cm}}{b\text{ m}}$
Solve the proportion. Multiply both parts of the scale by 4, because 2 times 4 is 8. Multiplying both parts of a ratio by the same positive number gives an equal ratio, so b is 5 times 4.
$\frac{2 \times 4}{5 \times 4} = \frac{8}{20}, \quad b = 20$
Let r be the actual length of the right side, in meters. Use the same kind of proportion with the 6 cm drawing length. Multiply both parts of the scale by 3, because 2 times 3 is 6.
$\frac{2\text{ cm}}{5\text{ m}} = \frac{6\text{ cm}}{r\text{ m}}, \quad \frac{2 \times 3}{5 \times 3} = \frac{6}{15}, \quad r = 15$
Let s be the actual length of the slanted side, in meters. The scale holds for this side too, because it and the real slanted side are corresponding sides. Multiply both parts of the scale by 5, because 2 times 5 is 10.
$\frac{2\text{ cm}}{5\text{ m}} = \frac{10\text{ cm}}{s\text{ m}}, \quad \frac{2 \times 5}{5 \times 5} = \frac{10}{25}, \quad s = 25$
Check with the scale factor. There are 100 cm in each meter, so 5 m is 500 cm, and the scale factor from the drawing to the park is 500 divided by 2, which is 250. Multiply each drawing length in centimeters by 250, then divide by 100 to change centimeters to meters. All three lengths agree.
$8 \times 250 = 2000\text{ cm} = 20\text{ m}, \quad 6 \times 250 = 1500\text{ cm} = 15\text{ m}, \quad 10 \times 250 = 2500\text{ cm} = 25\text{ m}$

Answer

The actual park has sides of $20$ m, $15$ m and $25$ m, and the $25$ m side is the slanted one.

Here is the idea to take with you. A scale compares a drawing length to the actual length it stands for, and the same scale holds for every pair of corresponding sides. Write both parts in the same unit, and the scale gives you a scale factor, a single positive number with no unit. Multiply a drawing length by the scale factor from the drawing to the actual object, and you get the corresponding actual length in that same unit. A proportion built from the scale gives the same result. All of this is for lengths; areas change by the square of the scale factor.

Next, you will find missing lengths in scale drawings of polygons and solve scale drawing word problems.