7th Grade · 7.G.A.1

Scale drawings in 7th grade

Scale drawings are drawings of real objects in which every length is shrunk or enlarged by the same amount, like a floor plan or a map. In 7th grade, your child uses the scale to find actual lengths and areas. Your child also redraws a scale drawing at a different scale.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

What it means, in plain words

The scale says how lengths on the drawing match lengths in real life. It can be written in words, such as "1 inch represents 4 feet". It can be a ratio, 1:48, because 4 feet is 48 inches. It can also be a bar printed on a map. All three say that one unit on the drawing stands for a fixed amount in real life.

A scale factor is the number that every length is multiplied by. If a copy is 3 times the size of the original, the scale factor is 3. A scale factor less than 1, such as $\frac{1}{2}$ , makes a smaller copy. To find an actual length, your child multiplies the drawing length by the scale and keeps track of the units.

Areas work differently from lengths in scale drawings. If every length is multiplied by 4, the area is multiplied by $4 \times 4 = 16$ , because the length and the width both grow. To redraw a scale drawing at a new scale, your child usually finds the actual lengths first, then turns them into lengths at the new scale.

Your child should be able to:

Read a scale given in words, as a ratio or as a bar
Find actual lengths from a scale drawing, and drawing lengths from actual ones
Find an actual area from a scale drawing
Redraw a scale drawing at a different scale

Example

On a floor plan, 1 inch represents 4 feet. A bedroom measures 3 inches by 2.5 inches on the plan. What are the real length, width and area of the room?

Length: $3 \times 4 = 12$ feet.
Width: $2.5 \times 4 = 10$ feet.
Area: $12 \times 10 = 120$ square feet.
Another way: the plan area is $3 \times 2.5 = 7.5$ square inches, and $7.5 \times 16 = 120$ . The room is 12 feet by 10 feet, with an area of 120 square feet.

Example

A garden is 9 centimeters long on a drawing where 1 centimeter represents 2 meters. How long is it on a new drawing where 1 centimeter represents 3 meters?

Actual length: $9 \times 2 = 18$ meters.
New drawing: $18 \div 3 = 6$ centimeters.
The garden is 6 centimeters long on the new drawing.

Where children usually get stuck

A patio is 5 inches by 3 inches on a drawing where 1 inch represents 10 feet. Your child multiplies the drawing area, 15 square inches, by 10 and says the patio covers 150 square feet.
Your child scales the area like a length. The area grows in two directions, so it is multiplied by the scale twice: $10 \times 10 = 100$ .
What helps. Have your child find the real length and width first, 50 feet and 30 feet, and multiply those. The patio covers 1,500 square feet.
With a scale of 1 inch to 5 feet, your child writes the ratio as 1:5.
The two numbers are in different units, so 1:5 compares inches with feet. In the same unit, 5 feet is 60 inches, so the ratio is 1:60.
What helps. Ask "1 what to 5 what?" every time your child writes a ratio from a scale.
To enlarge a 4 by 6 rectangle so the long side is 9, your child adds 3 to both sides and draws a 7 by 9 rectangle.
Adding the same amount to each side feels fair, but it changes the shape. A scale copy multiplies every length by the same number, here 1.5, so the copy is 6 by 9.
What helps. Draw both rectangles on graph paper next to the original. The 7 by 9 one looks squarer, while the 6 by 9 one keeps the shape.

Try four practice questions

Real 7th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Look at the original figure and its scale drawing. What is the scale factor from the original figure to the scale drawing?
Original figure
Scale drawing
Enter the scale factor.
Loading...
Show the answer and how to solve it
Answer. 2
The scale factor is $2$ because the corresponding side lengths change from $6$ cm to $12$ cm, and $\frac{12}{6} = 2$ .
1. 1Look at the matching labeled sides on the two rectangles.
2. 2The original figure has a labeled side of 6 cm and the scale drawing has the corresponding side labeled 12 cm. Divide the drawing's measurement by the original's.
3. 3The scale factor is $2$ .
Common mistake. A common mistake is writing $\frac{1}{2}$ by reversing the order of the comparison. The question asks for original figure to scale drawing, so use $\frac{12}{6}$ , not $\frac{6}{12}$ .
The idea underneath. A scale factor is the single multiplier that changes each original side length into the matching side length in the drawing, so use drawing length divided by original length for corresponding sides.
2On level
A segment is 6 cm on a scale drawing. Marco wrote the work below to find the actual length. Which statement best describes his mistake?
$\displaystyle \text{Scale: }1\text{ cm on the drawing} = 5\text{ cm on the actual figure}$
$\displaystyle \text{actual length} = 6\text{ cm}$
A
The student used a division setup instead of a multiplication setup.
B
The student treated the drawing length as the actual length.
C
The student squared the scale factor.
D
The student used the scale backwards and multiplied by $\frac{1}{5}$ instead of 5.
Show the answer and how to solve it
The correct statement is that the student treated the drawing length as the actual length. Since 1 cm on the drawing stands for 5 cm on the actual figure, 6 cm should become 30 cm, not 6 cm.
1. 1Read the scale first. It says 1 cm on the drawing equals 5 cm on the actual figure, so each drawing centimeter should be multiplied by 5.
2. 2A 6 cm drawing length should be multiplied by 5 to get the actual length of 30 cm.
3. 3The student's work just states the actual length is 6 cm — the same as the drawing length. The scale factor of 5 was never applied. The best description is that the student treated the drawing length as the actual length.
Common mistake. A common mistake is writing the drawing length directly as the actual length without applying the scale. Since the scale factor is 5, the actual length is $6 \times 5 = 30$ cm, not 6 cm.
The idea underneath. When a scale says 1 drawing unit equals n actual units with n > 1, the actual length must be n times larger, so a correct setup makes the number grow, not shrink.
3On level
The ratio below compares drawing lengths to actual lengths. Which expression finds the actual area?
$\displaystyle \text{Drawing area} = 12\text{ square units} \quad \text{drawing:actual} = 1:2$
A
$\displaystyle 12\times 2^2$
B
$\displaystyle 12\times\left(\frac{1}{2}\right)^2$
C
$\displaystyle \left(12\times 2\right)^2$
D
$\displaystyle 12\times 2$
Show the answer and how to solve it
The correct expression is $12 \times 2^{2}$ , which gives an actual area of 48 square units, because the actual-to-drawing length factor is 2 and area uses that factor squared.
1. 1The drawing area is 12 square units and the labeled ratio is drawing:actual = 1:2.
2. 2Because drawing:actual = $1:2$ , the actual-to-drawing length factor is $\frac{2}{1} = 2$ .
3. 3Area scales by the square of the length factor, so the correct expression is $12 \times 2^{2} = 48$ square units.
Common mistake. A common mistake is choosing $12 \times 2$ . That uses the length scale factor only once, but area changes in 2 dimensions, so the factor must be squared.
The idea underneath. For a rectangle, area is length $\times$ width. Both lengths are multiplied by $2$ , so the area is multiplied by $2 \times 2 = 2^2$ . The same rule holds for any scaled figure.
4Stretch
Use the scale and drawing length shown below. What is the actual length?
$\displaystyle \text{Scale: }2\text{ in} : 3\text{ yd} \qquad \text{Length on drawing: }8\text{ in}$
Actual length:ft
Show the answer and how to solve it
Answer. 36
The actual length is 36 ft because 8 in is 4 groups of 2 in, each group matches 3 yd, giving 12 yd, which equals 36 ft.
1. 1The scale says 2 in on the drawing matches 3 yd in real life. The drawing length is 8 in, and the answer must be in ft.
2. 2Find how many groups of 2 in are in 8 in.
3. 3Each group represents 3 yd, so multiply 4 by 3 to get the actual length in yards.
4. 4Convert yards to feet using 1 yd = 3 ft.
5. 5The actual length is 36 ft.
Common mistake. A common mistake is stopping at $12$ yd without converting to feet. The answer input shows ft, so you must multiply by 3 because $1$ yd $= 3$ ft.
The idea underneath. A scale is multiplicative: once you know how many scale groups fit in the drawing, use the same factor on the actual side.