Scale Drawings Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Scale Drawings.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Creating or interpreting drawings and models where every length is multiplied by the same constant (the scale factor), preserving shape while changing size.

A map is a scale drawing of the real world. If 1 inch on the map equals 10 miles in reality, the scale factor is 1:10 miles1:10\text{ miles}. Every distance on the map uses the same ratio, so the shapes stay accurate—just smaller. Enlarging a photo works the same way in reverse.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A scale drawing multiplies every real length by the same scale factor, keeping shape while changing size.

Common stuck point: The procedure for scale drawings is the easy part; the trap is multiplying area by the linear scale factor. Asking "Is every length related to the real object by one fixed multiplier (the scale factor)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is every length related to the real object by one fixed multiplier (the scale factor)?

Worked Examples

Example 1

easy
A map has a scale of 11 cm :50: 50 km. Two cities are 3.53.5 cm apart on the map. What is the actual distance between the cities?

Answer

175175 km

First step

1
Step 1: Identify the scale factor: 11 cm on the map =50= 50 km in reality.

Full solution

  1. 2
    Step 2: Write the proportion: 1 cm50 km=3.5 cmx km\frac{1 \text{ cm}}{50 \text{ km}} = \frac{3.5 \text{ cm}}{x \text{ km}}.
  2. 3
    Step 3: Solve by cross-multiplying: x=3.5×50=175x = 3.5 \times 50 = 175 km.
  3. 4
    Step 4: The actual distance between the two cities is 175175 km.
Scale drawings use proportional reasoning. The scale ratio 1:50 means every 1 cm on the map represents 50 km in reality. Multiplying the map distance (3.5 cm) by the scale factor (50) gives the actual distance of 175 km.

Example 2

medium
An architect draws a floor plan with a scale of 14\frac{1}{4} inch =1= 1 foot. A room measures 3.53.5 inches by 22 inches on the drawing. What are the actual dimensions and area of the room?

Example 3

medium
An architect's drawing has scale 12 inch=1 foot\tfrac{1}{2}\text{ inch} = 1\text{ foot}. A wall is drawn 77 inches long. How long is the real wall?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A model car is built at a scale of 1:241:24. The model car is 1515 cm long. How long is the actual car in meters?

Example 2

hard
A park is drawn on a map using a scale of 22 cm :300: 300 m. On the map, the park is a rectangle 55 cm by 33 cm. Find the actual area of the park in square meters and in hectares (1 hectare = 10,000 m²).

Example 3

easy
A map uses scale 1 cm:20 km1\text{ cm} : 20\text{ km}. A river is 77 cm long on the map. How long is it in reality?

Example 4

easy
A model bridge at scale 1:801:80 is 2525 cm long. How long is the real bridge in meters?

Example 5

easy
A real garden is 4040 m wide. On a scale 1:5001:500 map, how wide is it?

Example 6

easy
A map scale is 1:50,0001:50{,}000. Two landmarks are 44 cm apart. How far apart are they in km?

Example 7

easy
Scaling lengths by factor k=3k = 3 multiplies the area by what factor?

Example 8

medium
A floor plan at scale 1:401:40 shows a room 55 cm ×\times 88 cm. Find the room's real area in m2\text{m}^2.

Example 9

medium
A scale model of a building is at 1:751:75. The model is 4040 cm tall. How tall is the real building in meters?

Example 10

medium
A rectangular park is 300300 m by 500500 m. Drawn on a map at scale 1:10,0001:10{,}000, what are the map dimensions in cm?

Example 11

medium
A scale drawing has scale 1:251:25. The real area of a plot is 625625 m2^2. What is the drawing's area in cm2^2?

Example 12

medium
Two maps cover the same region: scale 1:25,0001:25{,}000 and 1:100,0001:100{,}000. A road is 88 cm long on the more detailed map. How long is it on the less detailed map?

Example 13

medium
A toy train is at scale 1:871:87. The real locomotive is 2020 m long. How long is the model in cm?

Example 14

medium
A 1:201:20 model boat holds 0.50.5 L of water. How much does the real boat hold (assuming volume scaling)?

Example 15

hard
A lake covers 3.23.2 km2^2 in reality and 2020 cm2^2 on a map. Find the linear scale 1:n1:n.

Example 16

hard
Two similar triangles have areas 3232 and 5050. Find the ratio of their corresponding sides.

Example 17

hard
Two similar prisms have volumes 5454 cm3^3 and 128128 cm3^3. Find the ratio of their surface areas.

Example 18

hard
A blueprint shows a rectangle 44 cm ×\times 33 cm at scale 1:501:50. The real rectangle's diagonal is how long?

Example 19

hard
A 1:301:30 model truck has a fuel tank that holds 5050 mL of liquid. Estimate how many liters the real truck's tank holds.

Example 20

hard
A scale model of a sculpture (scale 1:61:6) weighs 44 kg and is made of the same material as the original. Find the original's mass.

Example 21

challenge
An architect draws a 3030 m tall tower on A4 paper that is 29.729.7 cm tall, allowing 33 cm of margin at the top and bottom. What is the largest sensible scale 1:n1:n (smallest nn) usable, with nn a round multiple of 5050?

Example 22

challenge
A 1:251:25 model of a swimming pool holds 3232 L of water. The real pool's owner wants to know how many cubic meters fill it.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

ratiosproportionsmultiplicationsimilarity