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Scale Factors, Scale Drawings, and Maps

Use a scale or scale factor to go back and forth between a drawing, map, or model and the real thing.

Practice this skill 22 min Scale factors and scale drawings Leer en español
Key idea

What you'll learn

Maps, blueprints, floor plans, model cars, and enlarged photos all have one thing in common: every length is multiplied by the same number. That number is the scale factor, and the rule that connects the drawing to real life is the scale.

In this lesson you will learn to:

  • find a scale factor and tell whether a figure was enlarged or reduced;
  • use a map or drawing scale to find a real distance (and the reverse);
  • change a scale like "\tfrac{1}{4} inch = 1 foot" into a pure ratio like 1:48;
  • find missing sides of similar figures;
  • handle the GED-hard twist: areas scale by the square of the scale factor.

Why it matters on the GED: Expect questions with a map and a scale bar, a floor plan with a scale, a model built "at a scale of 1:24," or two similar triangles where you must find a missing side. These often come with a figure, and the trap answers come from dividing when you should multiply or forgetting to convert units.

Why it matters in life: Reading a road map, checking whether a couch fits on a floor plan, or ordering a photo enlargement all use scale.

Words to know

  • Scale drawing — a drawing of a real object where every length is shrunk (or enlarged) by the same amount. A floor plan is a scale drawing.
  • Scale — the rule connecting drawing lengths to real lengths, like "1 inch = 20 miles" or 1:50.
  • Scale bar — a small ruler printed on a map that shows the scale, such as a 1-inch bar labeled "12 miles."
  • Scale factor (k) — the number you multiply every original length by to get the new length. k = \dfrac{\text{new length}}{\text{original length}}. A 4-in photo enlarged to 10 in has k = 2.5.
  • Enlargement — a copy that is bigger than the original (k > 1).
  • Reduction — a copy that is smaller than the original (0 < k < 1). Halving every side means k = \tfrac{1}{2}.
  • Similar figures — figures with the same shape but different sizes: equal angles and proportional sides.
  • Corresponding sides — sides in the same position in two similar figures (shortest with shortest, longest with longest).
  • Ratio scale — a scale with no units, like 1:24, meaning 1 unit on the model = 24 of the same unit in real life.

All Math words in the glossary

Reading a scale: drawing ↔ real life

A scale such as 1 inch = 15 feet is a unit rate: each inch on the drawing stands for 15 real feet.

Drawing → real life: multiply.

\text{real length} = \text{drawing length} \times \text{real units per drawing unit}
  • A wall is 4 in long on the plan: 4 \times 15 = 60 ft.
  • A hallway is 2.5 in: 2.5 \times 15 = 37.5 ft.

Real life → drawing: divide.

\text{drawing length} = \frac{\text{real length}}{\text{real units per drawing unit}}
  • A 45-ft room on the plan: 45 \div 15 = 3 in.

When the scale has a fraction. Blueprints often say "\tfrac{1}{4} inch = 1 foot." Turn it into "1 inch = ? feet" first: 1 inch holds four quarter-inches, so 1 inch = 4 feet. A 3\tfrac{1}{2}-in wall is then 3.5 \times 4 = 14 ft.

Setting up a proportion. You can always write the scale and the unknown as two equal ratios, keeping the same units on top:

\frac{1\text{ in}}{15\text{ ft}} = \frac{4.2\text{ in}}{x\text{ ft}} \quad\Rightarrow\quad x = 4.2 \times 15 = 63\text{ ft}

Map scale bars. If the bar shows 1 inch = 12 miles and two towns are 2.25 inches apart, the real distance is 2.25 \times 12 = 27 miles. For a trip with several legs, add the map lengths first and then multiply (or multiply each leg and add — you get the same total).

Map drawn to scale with 1 inch = 12 miles. Town A and Town B are 2.25 inches apart on the map. A ruler under the map marks 0, 1, 2 and 3 inches as 0, 12, 24 and 36 miles; the 2.25-inch mark is 2.25 × 12 = 27 miles.Scale: 1 inch = 12 milesTown ATown B2.25 in on the map0 in0 mi1 in12 mi2 in24 mi3 in36 mi2.25 × 12 = 27 mi

Scale factors, enlargements, and ratio scales

Finding the scale factor. Divide a new length by the matching original length:

k = \frac{\text{new}}{\text{original}}
OriginalNewkType
4 in10 in10 \div 4 = 2.5enlargement
12 ft3 ft3 \div 12 = 0.25reduction
6 cm6 cm1same size

Every length in the figure gets multiplied by the same k. If a 4-by-6 photo is enlarged with k = 2.5, it becomes 10 by 15.

A 4 inch by 6 inch photo and its enlargement drawn to the same scale. Every side is multiplied by the scale factor k = 2.5, so the enlargement is 10 inches by 15 inches.Photo enlarged with k = 2.54 in6 inoriginal10 in15 inenlargement4 × 2.5 = 106 × 2.5 = 15× 2.5

Working backwards. If you know the new length and k, divide: original = \text{new} \div k. A copy made at k = 0.75 is 9 in wide, so the original was 9 \div 0.75 = 12 in.

Ratio scales have no units. A model car built at 1:24 means every 1 inch of model is 24 inches of real car (or 1 cm is 24 cm). The units are the same on both sides, so you must convert before using it.

  • A real truck is 18 ft long. In inches: 18 \times 12 = 216 in. Model: 216 \div 24 = 9 in.

Turning a unit scale into a ratio. "\tfrac{1}{4} in = 1 ft" means \tfrac{1}{4} in : 12 in. Multiply both sides by 4: 1 : 48. So a 1:48 blueprint is the same as "\tfrac{1}{4} inch = 1 foot."

Similar figures and what happens to area

Similar figures have proportional corresponding sides. To find a missing side:

  1. Match corresponding sides (shortest to shortest, longest to longest).
  2. Find k from a pair where you know both lengths.
  3. Multiply (or divide) the side you need.

Example. Triangle 1 has sides 6, 8, 10. A similar Triangle 2 has shortest side 15. Then k = 15 \div 6 = 2.5, and the side matching 8 is 8 \times 2.5 = 20.

Two similar right triangles drawn to scale. Triangle 1 has legs 6 and 8 and hypotenuse 10. Triangle 2 has shortest side 15, so the scale factor is 15 divided by 6 = 2.5; the side matching 8 is 8 × 2.5 = 20 and the hypotenuse is 25.Similar triangles: k = 15 ÷ 6 = 2.58610Triangle 18 × 2.5 = 201525Triangle 2

Perimeter scales by k. Perimeter is a length, so it is multiplied by k too. Triangle 1's perimeter is 24; Triangle 2's is 24 \times 2.5 = 60.

Area scales by k^2 — the GED-hard part. Area uses two lengths multiplied together, and each one is multiplied by k:

\text{new area} = k^2 \times \text{original area}
Scale factor kLengthsArea
2× 2× 4
3× 3× 9
\tfrac{1}{2}× \tfrac{1}{2}× \tfrac{1}{4}

Example. A 3-by-5 rectangle (area 15) is enlarged with k = 2. New sides 6 and 10, new area 60 = 4 \times 15 — not 30.

A 3 by 5 rectangle with area 15, and its enlargement with scale factor 2, a 6 by 10 rectangle. The enlargement contains 4 copies of the original, so its area is 60 = 4 × 15, not 30.Double the sides → 4 times the area53area 15original15151515106k = 2: area 60 = 4 × 15 (not 30)

Real area from a scale drawing. The safest method is to convert each length to real life first, then multiply. A 6 in by 4 in plan at 1 in = 10 ft is a 60 ft by 40 ft room: 60 \times 40 = 2{,}400 sq ft. (Using the drawing area 24 sq in \times\,10 would give a wrong 240.)

Working backwards from area. If the new area is 9 times the old, then k^2 = 9 and k = \sqrt{9} = 3.

Formula

Formulas to know

k = \frac{\text{new length}}{\text{original length}} \qquad \text{real} = \text{drawing} \times \text{scale rate} \qquad \text{drawing} = \frac{\text{real}}{\text{scale rate}} \text{new perimeter} = k \times \text{old perimeter} \qquad \text{new area} = k^2 \times \text{old area}
  • k — scale factor (k > 1 enlarges, k < 1 reduces).
  • scale rate — real units per one drawing unit (e.g. 15 ft per inch).

On the GED formula sheet? No. None of these scale formulas is on the sheet — you must remember them. The sheet does give area formulas for rectangles, triangles, and circles, which you may need once you have the real lengths. Unit conversions (12 in = 1 ft) are usually given in the question when needed, but know the common ones.

Floor plan of a rectangular living room drawn to scale. On the drawing the room is 5 inches long and 3.5 inches wide. The scale is 1 inch = 4 feet.Living room5 in3.5 inScale: 1 inch = 4 feet

A typical GED floor plan. Every inch on the drawing stands for 4 real feet, so the 5-in side is 5 \times 4 = 20 ft long.

Worked example

Example 1 (easy): map distance

On a map, 1 inch represents 15 miles. Two cities are 4.25 inches apart on the map. How far apart are they really?

  1. Map → real life, so multiply by the miles per inch.
  2. 4.25 \times 15 = 63.75
Map drawn to scale with 1 inch = 15 miles. Two cities are 4.25 inches apart. A ruler marks 0 to 5 inches as 0, 15, 30, 45, 60 and 75 miles; the cities sit at 4.25 inches, which is 4.25 × 15 = 63.75 miles.Scale: 1 inch = 15 milesCity 1City 24.25 in on the map0 in0 mi1 in15 mi2 in30 mi3 in45 mi4 in60 mi5 in75 mi4.25 × 15 = 63.75 mi

Answer: 63.75 miles.

Check: 4 inches would be 60 miles and a quarter inch is 15 \div 4 = 3.75 miles; 60 + 3.75 = 63.75 ✓

Worked example

Example 2: real life → drawing

A landscape plan uses the scale 1.5 in = 12 ft. A patio is 21 ft long. How long should it be on the plan?

  1. Find the scale per 1 inch: 12 \div 1.5 = 8, so 1 in = 8 ft.
  2. Real → drawing, so divide: 21 \div 8 = 2.625
Double number line for the landscape plan, drawn to scale: 0 inches = 0 feet, 1 inch = 8 feet, 1.5 inches = 12 feet, and 2.625 inches = 21 feet, the length of the patio.Plan scale 1.5 in = 12 ft, so 1 in = 8 ftplan (in)real (ft)00181.5122.62521

Answer: 2.625 inches (that is, 2\tfrac{5}{8} in).

Check: 2.625 \times 8 = 21 ft ✓

Worked example

Example 3 (work backwards): find the scale factor

A copy machine reduces a poster. A 12-inch side becomes 8 inches. What is the scale factor, and how long does a 15-inch side become?

  1. k = \frac{\text{new}}{\text{original}} = \frac{8}{12} = \frac{2}{3}
  2. k < 1, so this is a reduction.
  3. New length of the 15-in side: 15 \times \tfrac{2}{3} = 10\text{ in}
A 12 inch by 15 inch poster and its reduced copy drawn to scale. The 12-inch side becomes 8 inches, so k = 8/12 = 2/3, and the 15-inch side becomes 15 × 2/3 = 10 inches.Reduction with k = 8 ÷ 12 = 2/312 in15 inposter8 in10 incopy15 × 2/3 = 10× 2/3

Answer: k = \tfrac{2}{3}; the 15-in side becomes 10 in.

Check: \tfrac{10}{15} = \tfrac{2}{3}, the same ratio as \tfrac{8}{12} ✓

Worked example

Example 4: a ratio scale with unit conversion

An architect's model uses the scale 1:48. A real building is 33 ft wide. How wide is the model, in inches?

  1. A ratio scale has the same units on both sides, so convert feet to inches: 33 \times 12 = 396\text{ in}
  2. Divide by 48: 396 \div 48 = 8.25\text{ in}
Tape diagram for the 1 to 48 scale, drawn to scale. One small box is the model width, 8.25 inches. The real building width is a row of 48 such boxes: 48 × 8.25 = 396 inches, which is 33 feet.Scale 1:48 — the real width is 48 model widthsmodel: 8.25 inbuilding: 48 × 8.25 in = 396 in = 33 ft48 equal pieces

Answer: 8.25 inches.

Check: 8.25 \times 48 = 396 in, and 396 \div 12 = 33 ft ✓

Worked example

Example 5 (GED-hard): area of an enlargement

A logo has an area of 18 square inches. It is enlarged so that every length is 2.5 times as long. What is the area of the enlarged logo?

  1. Lengths scale by k = 2.5, so area scales by k^2.
  2. k^2 = 2.5^2 = 6.25
  3. \text{new area} = 6.25 \times 18 = 112.5
The logo pictured as a 3 inch by 6 inch rectangle with area 18 square inches, and its enlargement with scale factor 2.5: a 7.5 inch by 15 inch rectangle with area 112.5 square inches, which is 6.25 times 18.k = 2.5, so area × 2.5² = × 6.256 in3 inarea 18 sq in15 in7.5 inarea 112.5 sq in6.25 × 18 = 112.5

Answer: 112.5 square inches.

Check: Picture the logo as a 3 in × 6 in rectangle (area 18). Enlarged: 7.5 in × 15 in, and 7.5 \times 15 = 112.5 ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

  • Scales with fractions: use n/d to enter a fraction such as \tfrac{1}{4} (type 1, press the down arrow, type 4, then the right arrow to leave the fraction). For a mixed number like 3\tfrac{3}{4}, the simplest entry is the decimal 3.75. Press toggle to switch a fraction answer to a decimal.
  • Area factor: square the scale factor with the x² key: 2.5 x² gives 6.25.
  • Working backwards from area: take the square root with 2nd + x² (√). If the area grew from 24 to 216, compute 216 \div 24 = 9, then √9 = 3.
  • When not to use it: most map questions use friendly numbers (3 \times 20, 45 \div 15). Do those in your head — and estimate to check any calculator result.
Common mistake

Common mistakes

  • Dividing when you should multiply. 3.5 in at 1 in = 20 mi is not 20 \div 3.5 \approx 5.7 mi. ✔ Drawing → real: multiply (3.5 \times 20 = 70 mi).
  • Flipping the scale factor. A 4-in photo becomes 10 in. 4 \div 10 = 0.4 would shrink it. ✔ k = \text{new} \div \text{original} = 2.5.
  • Ignoring units in a ratio scale. At 1:24, a 15-ft car is not 15 \div 24 = 0.625 "inches." ✔ Convert to inches first: 180 \div 24 = 7.5 in.
  • Scaling area by k instead of k^2. Doubling the sides of a 15-sq-in rectangle does not give 30 sq in. ✔ It gives 2^2 \times 15 = 60 sq in.
  • Multiplying a drawing area by the length scale. 24 sq in at 1 in = 10 ft is not 240 sq ft. ✔ Convert each side, then multiply: 60 \times 40 = 2{,}400 sq ft.
  • Matching the wrong sides in similar figures. ✔ Shortest to shortest, longest to longest — and use the picture's labels.
Test strategy

GED test strategy

  • Ask "bigger or smaller?" Real roads are longer than map lines; models are smaller than cars. Eliminate any answer that goes the wrong direction.
  • Write the scale as "1 drawing unit = ___ real units" before doing anything else.
  • Circle the units in the question and in the answer choices. If the choices are in inches and the scale is in feet, a conversion is coming.
  • Plug back in: multiply your drawing answer by the scale and see if you get the real length.
  • Area question? Look for the words area, square feet, square inches, or covers. That's your cue to square k — or convert sides first.
  • Use the figure. Scale-drawing figures on the GED are labeled; read every label, and use the scale bar printed on the map.

Check yourself

Scale factors and scale drawingsNo calculator

On a state map, 1 inch represents 20 miles. Two towns are 3.5 inches apart on the map. What is the actual distance between the towns?

Answer choices
Hint

Each inch on the map stands for 20 real miles. Should the real distance be more or less than 3.5?

Check yourself

Scale factors and scale drawingsNo calculatorFill in

A photo that is 4 inches wide is enlarged so that it is 10 inches wide. What is the scale factor of the enlargement?

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Scale factor = new length ÷ original length.

Summary

Key points

  • A scale is a unit rate: drawing → real, multiply; real → drawing, divide.
  • The scale factor is k = \text{new} \div \text{original}; k > 1 enlarges, k < 1 reduces.
  • A ratio scale (like 1:48) has the same units on both sides — convert before using it.
  • Similar figures: match corresponding sides, find k, then multiply or divide.
  • Lengths and perimeters scale by k; areas scale by k^2.
  • Scale formulas are not on the GED formula sheet — memorize them.
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