Actual Areas and Redrawing at a New Scale | Grade 7 Math | LevelKite

A scale drawing shows a real object with every length shrunk or enlarged by the same factor. You already know how to use its scale to turn a drawing length into an actual length, the length on the real object. A scale compares a drawing length with the actual length of the corresponding side, the side of the real object that the drawing side stands for. A scale of $1$ cm : $3$ m means every $1$ cm on the drawing represents $3$ m on the real object.

Keep the units consistent: with that scale, measure every drawing length in centimeters, and every actual length comes out in meters.

This lesson answers two new questions:

How large is the actual area, the area of the real object?
How do you redraw the same object at a different scale?

Start with one grid square. In the figure below, the horizontal axis is $x$ and the vertical axis is $y$ , and both count centimeters on the drawing. Each grid square is $1$ cm on each side, and the scale is $1$ cm : $3$ m.

One grid square has a drawing area of $1 \text{ cm} \times 1 \text{ cm} = 1 \text{ cm}^2$ . The real square it stands for is $3$ m long and $3$ m wide, so it covers $3 \text{ m} \times 3 \text{ m} = 9 \text{ m}^2$ . Each square centimeter of the drawing stands for $9$ square meters, not $3$ .

The rectangle in the figure is a scale drawing of a garden. Its corners are $(1, 1)$ , $(5, 1)$ , $(5, 3)$ and $(1, 3)$ , and it covers $8$ grid squares. So you can predict the garden's area: $8$ squares, each standing for $9$ square meters.

Each grid square is

1

cm by

1

cm, and the scale is

1

cm :

3

m. The rectangle is

4

cm wide and

2

cm tall, so it covers

8

grid squares.

Worked example

Actual area of a rectangular garden

The rectangle in the figure is a scale drawing of a garden. Each grid square is $1$ cm on each side, and the scale is $1$ cm : $3$ m. The corners are $(1, 1)$ , $(5, 1)$ , $(5, 3)$ and $(1, 3)$ . What is the actual area of the garden?

Find the drawing lengths. The sides lie along grid lines, so each length is a difference of coordinates. A length is positive, so subtract the smaller coordinate from the larger one. Each grid unit is 1 cm, so the rectangle is 4 cm wide and 2 cm tall.
$5 - 1 = 4, \quad 3 - 1 = 2$
Change each drawing length to an actual length. The scale says every 1 cm represents 3 m, so multiply each number of centimeters by 3. The same scale applies to every side, so use it for both. The garden is 12 m long and 6 m wide.
$4 \text{ cm} \times 3 \text{ m per cm} = 12 \text{ m}, \quad 2 \text{ cm} \times 3 \text{ m per cm} = 6 \text{ m}$
Find the actual area. The area of a rectangle is its length times its width. Both actual lengths are in meters, so the area is in square meters.
$12 \text{ m} \times 6 \text{ m} = 72 \text{ m}^2$
Check by counting squares. The drawing's area is 4 cm times 2 cm, which is 8 square centimeters. Each square centimeter stands for a 3 m by 3 m square, which is 9 square meters.
$8 \text{ cm}^2 \times 9 \text{ m}^2 \text{ per cm}^2 = 72 \text{ m}^2$

Answer

The actual area of the garden is $72$ square meters.

Key concept

Lengths use the scale factor; areas use its square

Let $k$ be the scale factor: the positive number of actual units that $1$ drawing unit represents. For the garden, $k = 3$ , because $1$ cm represents $3$ m.

A scale compares corresponding sides, and the units must stay consistent: every drawing length in one unit and every actual length in one unit. Then:

$\text{actual length} = k \times \text{drawing length}$
$\text{actual area} = k^2 \times \text{drawing area}$

Why $k^2$ ? Each $1$ -by- $1$ square on the drawing stands for a $k$ -by- $k$ square on the real object, and that square has area $k \times k$ . For the garden, $k^2 = 3 \times 3 = 9$ , so every $1 \text{ cm}^2$ stands for $9 \text{ m}^2$ .

\text{actual area} = k^2 \times \text{drawing area}

Each grid square is

1

cm by

1

cm, and the scale is

1 : 2000

. The horizontal side is

4

cm, the vertical side is

3

cm, and the slanted side is given as

5

cm. The first two meet at a right angle at

(1, 1)

.

Worked example

Actual area of a triangular park

The triangle in the figure is a scale drawing of a park. The grid works as before: $x$ runs across, $y$ runs up, and each grid square is $1$ cm on each side. The scale is $1 : 2000$ , and the corners are $(1, 1)$ , $(5, 1)$ and $(1, 4)$ . The slanted side is given as $5$ cm. What is the actual area of the park in square meters?

Find a base and its height on the drawing. The side from (1, 1) to (5, 1) is horizontal and the side from (1, 1) to (1, 4) is vertical, so they are perpendicular: they meet at a right angle. Use them as the base and the height. Each grid unit is 1 cm, so subtracting coordinates gives a base of 4 cm and a height of 3 cm. The area does not need the slanted side.
$5 - 1 = 4, \quad 4 - 1 = 3$
Find the drawing's area. The area of a triangle is one half times its base times its height, where the height is perpendicular to the base.
$\tfrac{1}{2} \times 4 \text{ cm} \times 3 \text{ cm} = 2 \text{ cm} \times 3 \text{ cm} = 6 \text{ cm}^2$
Find the scale factor in useful units. A scale of 1 : 2000 has no units written, so both lengths use the same unit: 1 cm on the drawing represents 2000 cm in the park. The question asks for square meters, so change 2000 cm to meters. Since 100 cm = 1 m, divide by 100. Every 1 cm represents 20 m, so k = 20 for drawing lengths in centimeters and actual lengths in meters.
$2000 \div 100 = 20$
Use the area rule. Areas are multiplied by k squared, because each square centimeter of the drawing stands for a 20 m by 20 m square. That square covers 400 square meters, so multiply the drawing's area by 400.
$20 \times 20 = 400, \quad 6 \text{ cm}^2 \times 400 \text{ m}^2 \text{ per cm}^2 = 2400 \text{ m}^2$
Check by scaling the lengths first. The actual base is 4 times 20, or 80 m, and the actual height is 3 times 20, or 60 m. A scale drawing keeps every angle of the real object, so the base and height are still perpendicular. The triangle formula gives the same area.
$\tfrac{1}{2} \times 80 \text{ m} \times 60 \text{ m} = 40 \text{ m} \times 60 \text{ m} = 2400 \text{ m}^2$

Answer

The actual area of the park is $2400$ square meters.

A drawing may be too small to read or too big for the page, so you redraw it. To reproduce a scale drawing at a different scale means to make a new drawing of the same object with a new scale.

The real object does not change, so its actual lengths do not change. That gives you a path through the actual lengths:

First, use the old scale to find each actual length.
Then, use the new scale to turn each actual length into a new drawing length. If $1$ cm represents $2$ m, then $\text{actual length} = 2 \times \text{drawing length}$ , so you undo that by dividing the number of meters by $2$ .

Both scales must use consistent units, such as centimeters on the drawing and meters on the real object. Every side is multiplied and then divided by the same two positive numbers, so all the sides change by one common factor. Keep every angle the same too, and the new drawing has the same shape as the old one.

The next figure uses the same axes and $1$ cm grid squares. Its left rectangle is the old garden drawing. Its right rectangle is the new drawing that the example builds.

Left: the old drawing,

4

cm by

2

cm at

1

cm :

3

m. Right: the new drawing,

6

cm by

3

cm at

1

cm :

2

m. Each grid square is

1

cm by

1

cm.

Worked example

Redrawing the garden at a new scale

The left rectangle in the figure is the garden drawing from the first example. Its corners are $(1, 1)$ , $(5, 1)$ , $(5, 3)$ and $(1, 3)$ , each grid square is $1$ cm on each side, and its scale is $1$ cm : $3$ m. Reproduce the drawing at the new scale $1$ cm : $2$ m on the same grid, with its lower left corner at $(6, 1)$ .

Read the old drawing lengths as differences of coordinates, since the sides lie along grid lines. Each grid unit is 1 cm, so the old rectangle is 4 cm wide and 2 cm tall.
$5 - 1 = 4, \quad 3 - 1 = 2$
Use the old scale to find the actual lengths. Every 1 cm represents 3 m, so multiply by 3. The garden is 12 m long and 6 m wide, and that stays true at any scale.
$4 \text{ cm} \times 3 \text{ m per cm} = 12 \text{ m}, \quad 2 \text{ cm} \times 3 \text{ m per cm} = 6 \text{ m}$
Use the new scale to find the new drawing lengths. Now every 1 cm represents 2 m, so the number of meters is 2 times the number of centimeters. Undo that by dividing each number of meters by 2. You can divide because 2 is not zero.
$12 \text{ m} \div 2 \text{ m per cm} = 6 \text{ cm}, \quad 6 \text{ m} \div 2 \text{ m per cm} = 3 \text{ cm}$
Draw the new rectangle. Each grid unit is 1 cm, so 6 cm is 6 units and 3 cm is 3 units. From (6, 1), go right 6 units to (12, 1), up 3 units to (12, 4), and left 6 units to (6, 4), then close the shape. The sides stay horizontal and vertical, so every corner is still a right angle.
$6 + 6 = 12, \quad 1 + 3 = 4$
Check the shape. Each old length was multiplied by 3 and then divided by 2, so each new length should be 3 divided by 2, or 1.5, times its old length. This comparison works because both scales pair centimeters with meters.
$3 \div 2 = 1.5, \quad 4 \text{ cm} \times 1.5 = 6 \text{ cm}, \quad 2 \text{ cm} \times 1.5 = 3 \text{ cm}$
Check the area. The new drawing covers 6 cm times 3 cm, which is 18 square centimeters. At the new scale, each square centimeter stands for a 2 m by 2 m square, which is 4 square meters. The result matches the 72 square meters from the first example.
$18 \text{ cm}^2 \times 4 \text{ m}^2 \text{ per cm}^2 = 72 \text{ m}^2$

Answer

At the scale $1$ cm : $2$ m, the garden is a $6$ cm by $3$ cm rectangle with corners $(6, 1)$ , $(12, 1)$ , $(12, 4)$ and $(6, 4)$ .

Will a new drawing be bigger or smaller than the old one? Compare how many meters each centimeter stands for. Going from $1$ cm : $3$ m to $1$ cm : $2$ m, each centimeter stands for fewer meters. You need more centimeters to show the same garden, so the new drawing is larger.

Now try a scale where each centimeter stands for more meters. At $1$ cm : $6$ m, the drawing lengths are $12 \div 6 = 2$ cm and $6 \div 6 = 1$ cm, so the drawing shrinks.

The table lists these three drawings of the same $12$ m by $6$ m garden, from the largest to the smallest. Squaring the meters per centimeter tells you what each square centimeter stands for: $2^2 = 4$ , $3^2 = 9$ and $6^2 = 36$ . In every row, drawing area times that number gives the actual area: $18 \times 4$ , $8 \times 9$ and $2 \times 36$ all equal $72$ . The drawing changes with the scale, but the garden's actual area does not.

One garden drawn at three scales

Scale	Drawing size	Drawing area	Each square centimeter stands for	Actual area
$1$ cm : $2$ m	$6$ cm by $3$ cm	$18 \text{ cm}^2$	$4 \text{ m}^2$	$72 \text{ m}^2$
$1$ cm : $3$ m	$4$ cm by $2$ cm	$8 \text{ cm}^2$	$9 \text{ m}^2$	$72 \text{ m}^2$
$1$ cm : $6$ m	$2$ cm by $1$ cm	$2 \text{ cm}^2$	$36 \text{ m}^2$	$72 \text{ m}^2$

Here is the big idea. A scale compares corresponding sides in consistent units. Going from a drawing to the real object, lengths are multiplied by the scale factor $k$ and areas are multiplied by $k^2$ , because each drawing square stands for a $k$ -by- $k$ square.

To reproduce a drawing at a different scale, remember that the real object stays the same. Use the old scale to find the actual lengths, use the new scale to find the new drawing lengths, and keep every angle the same.

Next, you will use these ideas to find and fix mistakes in scale drawing work.