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Maps, floor plans and models: real lengths and areas from a scale, and redrawing at a new scale.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you work with scale drawings: maps, floor plans and models. You read a scale written with units, as a ratio or as a scale bar, and use it as a constant of proportionality to go from the drawing to the real object and back. You find the scale factor between a figure and its copy, see why areas grow by the square of the scale factor, and redraw a figure at a new scale. These are the tools you need before measuring triangles, circles and angles in real places.
You know how to find a unit rate and a constant of proportionality, such as miles per hour or dollars per pound. You can multiply and divide by fractions and decimals, change feet to inches, and find the area of a rectangle. A scale drawing puts these ideas to work: every length on the drawing is proportional to the real length, so one number, the scale, links the two. In this lesson you read maps, floor plans and models, find real lengths and areas, find the scale factor between two figures, and redraw a picture at a different scale.
| Term | What it means |
|---|---|
| Scale drawing | A drawing in which every length is the same multiple of the real length. |
| Scale | The rule linking drawing lengths to real lengths, such as 1 in : 20 mi. |
| Ratio scale | A scale with no units, such as 1 : 24, where both lengths use the same unit. |
| Scale factor | The number that multiplies every length of one figure to give the matching length of a copy. |
| Enlargement | A copy with a scale factor greater than 1. |
| Reduction | A copy with a scale factor between 0 and 1. |
| Corresponding lengths | Lengths in the same position on the original and on the copy. |
| Scale bar | A marked line on a map showing how far a distance on the map is on the ground. |
In a scale drawing, every real length is the same multiple of its drawn length. If a map says 1 inch : 20 miles, then every inch on the map stands for 20 miles on the ground, so
$$\text{real distance} = 20 \times \text{map distance}$$
Four inches on the map is 80 miles, and half an inch is 10 miles. Going the other way, from real to drawing, you divide: a 50-mile road is $50 \div 20 = 2.5$ inches long on the map.
Because the same number multiplies every length, the drawing keeps the exact shape of the real thing: angles stay the same, and a room twice as long as it is wide is drawn twice as long as it is wide. Only the size changes. Areas behave differently from lengths, though, and that is where most mistakes happen: when every length is multiplied by $k$, every area is multiplied by $k^2$.
Another way: picture
A floor plan of a rectangular room drawn on grid paper, 3 squares by 2 squares, with a note that one square is 4 feet. The real room is 12 feet by 8 feet, and each small square on the paper covers 16 square feet of floor.
Another way: story
A class wants to paint a mural of the school on a wall. They photograph the school, draw a grid on the photo, and copy each small square into a much larger square on the wall. Every square grows by the same factor, so the mural keeps the school's shape.
Scales are written in several ways, and all of them mean the same kind of thing.
When the drawing number is not 1, such as 2 inches : 15 miles, find the unit rate first: one inch stands for $15 \div 2 = 7.5$ miles.
A scale works in both directions, and a table makes the direction clear:
| Map (in) | 1 | 2 | 3.5 | 6 |
|---|---|---|---|---|
| Real (mi) | 20 | 40 | 70 | 120 |
Going down the table, from map to real, you multiply by 20. Going up, from real to map, you divide by 20. A quick sense check tells you which: real objects are usually bigger than their drawings, so a real length should come out larger. Models of tiny things, such as a large model of an ant or a cell, are the exception, and then the drawing is the bigger one.
A ratio scale such as 1 : 24 compares two lengths in the same unit, so you may have to convert first. A real car 15 feet long is $15 \times 12 = 180$ inches long, so its 1 : 24 model is $180 \div 24 = 7.5$ inches long. If you divided 15 feet by 24 you would get 0.625 feet, which is also correct, but harder to measure with a ruler. Pick the unit you want in the answer and convert before you scale.
Scales with units can hide a conversion too. The scale $\tfrac{1}{4}$ inch = 1 foot means one whole inch on the plan stands for 4 feet, so a line $3\tfrac{1}{2}$ inches long is $3.5 \times 4 = 14$ feet.
When you copy a figure larger or smaller, the scale factor is the number each length is multiplied by. Find it by dividing a copy length by the matching original length:
$$\text{scale factor} = \frac{\text{copy length}}{\text{original length}}$$
A 4 by 6 inch photo enlarged to 10 by 15 inches has a scale factor of $10 \div 4 = 2.5$, and the other pair agrees: $15 \div 6 = 2.5$. If the pairs gave different answers, the copy would be stretched, not a true scale copy. A factor greater than 1 is an enlargement, and a factor between 0 and 1, such as $\tfrac{2}{3}$, is a reduction. A factor of exactly 1 gives a copy the same size.
Suppose a garden plan shows a bed 3 inches by 2 inches at 1 inch : 5 feet. The real bed is 15 feet by 10 feet, so its area is 150 square feet. The area on the plan is only 6 square inches, and $6 \times 5 = 30$, not 150. What went wrong? Both the length and the width were multiplied by 5, so the area was multiplied by $5 \times 5 = 25$, and $6 \times 25 = 150$.
You can see it on grid paper: one square inch of the plan stands for a square 5 feet on each side, which is 25 square feet. The safe method is always to turn each length into a real length first and only then multiply. Then the squaring happens by itself and cannot be forgotten.
Sometimes a drawing has to be made again at a new scale: a plan that is too small to read, or a poster that must fit a smaller page. The simplest way is to go through the real lengths. Use the old scale to find each real length, then use the new scale to find each new drawn length.
For example, a patio drawn 6 inches long at 1 inch : 2 feet is really 12 feet long. At a new scale of 1 inch : 3 feet it is drawn $12 \div 3 = 4$ inches long. You can also go straight from drawing to drawing: the new drawing is $\tfrac{2}{3}$ the size of the old one, because each inch used to mean 2 feet and now means 3.
Use these steps with any scale drawing.
How to check. Ask whether the size makes sense: a real room should be bigger than its plan, and a mountain on a map should be miles, not inches. Check a length by reversing the step: divide your real length by the scale and see if you get the drawn length back. Check an area by multiplying the drawn area by the square of the scale.
In the United States, architects usually draw house plans at the scale $\tfrac{1}{4}$ inch = 1 foot, which is the same as the ratio 1 : 48. A living room drawn $4\tfrac{1}{2}$ inches by $3\tfrac{3}{4}$ inches is really $4.5 \times 4 = 18$ feet by $3.75 \times 4 = 15$ feet, with an area of $18 \times 15 = 270$ square feet. If carpet costs 3 dollars a square foot, the family budgets 810 dollars. Builders use special rulers, called architect's scales, that are marked in feet at each common scale, so they can read real lengths straight off the plan without doing the multiplication.
Before carving Mount Rushmore, the sculptor Gutzon Borglum made plaster models at a scale of 1 inch to 1 foot, or 1 : 12. Workers measured a point on the model and multiplied by 12 to find where to carve on the mountain. Each face on the mountain is about 60 feet tall, so on the model each face was about $60 \div 12 = 5$ feet tall. A nose that measured 1 foot 8 inches, or 20 inches, on the model became $20 \times 12 = 240$ inches, which is 20 feet, on the mountain.
The most popular size of model train in the United States is HO scale, about 1 : 87. A real boxcar 50 feet long is $50 \times 12 = 600$ inches long, so the model is about $600 \div 87 \approx 6.9$ inches long. Hobbyists use the scale in the other direction too: a model building 4 inches tall stands for $4 \times 87 = 348$ inches, or 29 feet, about the height of a two-story house.
The most common mistake is multiplying an area by the scale instead of its square. At 1 inch : 10 feet, one square inch of drawing is 100 square feet, not 10.
The second is going the wrong direction: dividing when you should multiply. Ask whether the answer should be bigger or smaller than the number you started with.
The third is ignoring units in a ratio scale. At 1 : 24, a 15-foot car is not 15 divided by 24 inches long; the 15 feet must become 180 inches first.
The fourth is adding instead of multiplying. Making a picture 3 inches wider on every side is not a scale copy, because it changes the shape. A scale copy multiplies every length by the same number.
The last is measuring the scale factor from one pair of sides only. Check a second pair; if the two factors disagree, the copy is stretched.
Read the scale as a unit rate.
$1 \text{ in} \to 20 \text{ mi}$
Every inch on the map stands for 20 miles on the ground.
Measure the route on the map.
$\text{map distance} = 3.5 \text{ in}$
Measure along the road, not in a straight line, if the road bends.
Choose the direction.
$\text{map} \to \text{real: multiply}$
Real distances are larger than map distances.
Multiply by the scale.
$3.5 \times 20 = 70 \text{ mi}$
Three and a half groups of 20 miles.
Check by reversing.
$70 \div 20 = 3.5 \text{ in}$
Dividing the real distance by the scale gives back the map distance.
Read the drawn length and width.
$3 \text{ in by } 2.5 \text{ in}$
These are measured on the plan.
Scale the length.
$3 \times 4 = 12 \text{ ft}$
Each inch of plan stands for 4 feet.
Scale the width.
$2.5 \times 4 = 10 \text{ ft}$
The same scale applies to every length.
Multiply the real lengths for the area.
$12 \times 10 = 120 \text{ ft}^2$
Area must be found from real lengths.
Find the drawn area.
$3 \times 2.5 = 7.5 \text{ in}^2$
This is only used to check.
Check with the square of the scale.
$7.5 \times 4^2 = 7.5 \times 16 = 120$
One square inch of plan covers 4 by 4 feet, which is 16 square feet.
Write the first scale as a unit rate.
$1 \text{ in} \to 8 \text{ ft}$
The first poster uses 1 inch for every 8 feet.
Find the drawn length at that scale.
$94 \div 8 = 11.75 \text{ in}$
Real to drawing is a division.
Find the drawn width at that scale.
$50 \div 8 = 6.25 \text{ in}$
The same division applies to every length.
Notice the drawing is too big for the page.
$11.75 > 10$
The page is only 10 inches wide, so a new scale is needed.
Choose a new scale and redraw the length.
$94 \div 10 = 9.4 \text{ in}$
At 1 inch : 10 feet, each inch covers more court, so the drawing shrinks.
Redraw the width at the new scale.
$50 \div 10 = 5 \text{ in}$
The new poster fits on the page: 9.4 inches by 5 inches.
Check the shape is the same.
$\dfrac{11.75}{6.25} = 1.88 = \dfrac{9.4}{5}$
Both drawings have the same length-to-width ratio as the real court, $94 \div 50 = 1.88$.
Divide a copy length by the matching original length.
$10 \div 4 = 2.5$
The scale factor takes each original length to its copy.
Check with the other pair of sides.
$15 \div 6 = 2.5$
Both pairs agree, so the enlargement is a true scale copy.
Find how many times as large the area is.
A drawing uses a scale where 1 inch represents $12$ feet. To find a real length from a length measured on the drawing, what should you do?
A wall is $6$ inches long on a plan drawn at 1 inch : $4$ feet. Complete the worked solution to find its length on a new plan drawn at 1 inch : $2$ feet.
Find the real length by multiplying by the first scale.
$6 \times 4 =$ r $\text{ ft}$
Each inch of the old plan stands for $4$ feet.
Divide the real length by the new scale.
$\text{real length} \div 2 =$ n $\text{ in}$
On the new plan, every $2$ feet of wall takes up one inch.
Check the size makes sense.
$2 < 4 \Rightarrow \text{new drawing is longer}$
Fewer feet per inch means more inches are needed for the same wall.
On a road map, 2 inches represent $11$ miles. Two towns are $5$ inches apart on the map. How many miles does 1 inch represent, and how far apart are the towns?
1 inch: u miles. Distance: answer miles.
A house plan uses the scale $\tfrac{1}{4}$ inch $=$ 1 foot. A bedroom on the plan is $7.5$ inches long and $5$ inches wide. What are its real length and width?
Length: l ft. Width: w ft.
A rectangle $7$ cm by $4$ cm is enlarged by a scale factor of $3$. How many times as large is the new area, and what is it?
k times as large; new area answer square cm.
A rectangle $35$ cm by $42$ cm is reduced to a copy whose short side is $10$ cm. What is the scale factor, as a fraction, and how long is the copy's long side?
Scale factor: f. Long side: answer cm.
A model car kit is built at a scale of 1 : 24. The real car is $16$ feet long. How long is the model, in inches? The model is $3.5$ inches wide; how wide is the real car, in feet?
Model length: m inches. Real width: answer feet.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
On a town map, 1 inch represents $60$ feet. A rectangular park is drawn $7$ inches by $4$ inches. Find the park's real length, real width and real area.
l ft by w ft; area answer square feet.
You can use scale drawings. Without looking: at 1 inch : 6 feet, how long is a wall drawn $2\tfrac{1}{2}$ inches long, and how many square feet does one square inch of the plan stand for?
19. Your turn: a 4 by 6 inch photo enlarged to 10 by 15 inches, step 3
$2.5^2 = 6.25$
Area grows by the square of the factor: the areas are 24 and 150 square inches, and $150 \div 24 = 6.25$.