Scale Drawings

Scale drawings use proportions to represent objects accurately at different sizes!

WHAT IS A SCALE DRAWING? A representation of an object where all dimensions are proportional to the actual object.

Examples: - Maps (cities, countries) - Blueprints (houses, buildings) - Model cars, airplanes - Diagrams in science books

SCALE: The ratio that compares the drawing size to the actual size.

Written as: - Ratio: 1:100 - Fraction: 1/100 - Statement: "1 inch represents 100 feet" - "1 cm = 5 m"

SCALE FACTOR: The multiplier used to go from actual size to drawing size (or vice versa).

If scale is 1:50, scale factor = 1/50 (drawing is 1/50 of actual size)

TYPES OF SCALES:

Reduction Scale (most common): Drawing is SMALLER than actual object - Example: 1 cm = 10 m (1 cm represents 10 meters) - Scale factor < 1

Enlargement Scale: Drawing is LARGER than actual object - Example: 10 cm = 1 mm (microscope view) - Scale factor > 1

FINDING ACTUAL SIZE FROM DRAWING:

Formula: Actual = Drawing × Scale Factor

Example: On a map with scale 1 cm = 50 km, a distance measures 3 cm. Find actual distance.

Method 1: Set up proportion 1 cm / 50 km = 3 cm / x km x = 3 × 50 = 150 km

Method 2: Multiply by scale 3 cm × 50 km/cm = 150 km

FINDING DRAWING SIZE FROM ACTUAL:

Formula: Drawing = Actual ÷ Scale Factor (or Actual × reciprocal)

Example: A building is 120 feet tall. On a blueprint with scale 1 in = 20 ft, how tall is the drawing?

Method 1: Set up proportion 1 in / 20 ft = x in / 120 ft 20x = 120 x = 6 inches

Method 2: Divide 120 ft ÷ 20 ft/in = 6 inches

FINDING THE SCALE:

Given drawing size and actual size, find the scale.

Example: A 60-foot room is drawn as 3 inches. Find the scale.

Scale = Drawing : Actual = 3 in : 60 ft = 1 in : 20 ft Scale: 1 in = 20 ft

SCALE AND AREA:

IMPORTANT: When dimensions are scaled by factor k: - Linear measurements (length, width): multiply by k - Area: multiply by k² - Volume: multiply by k³

Example: Scale factor 1:10 (actual is 10× bigger) - If drawing length = 5 cm, actual = 5 × 10 = 50 cm - If drawing area = 20 cm², actual = 20 × 10² = 2,000 cm²

PERIMETER AND AREA IN SCALE DRAWINGS:

Example: A rectangle on a blueprint is 4 in × 6 in. Scale is 1 in = 8 ft.

Actual dimensions: Length: 4 in × 8 ft/in = 32 ft Width: 6 in × 8 ft/in = 48 ft

Perimeter: Drawing: 2(4 + 6) = 20 in Actual: 2(32 + 48) = 160 ft Notice: 20 in × 8 ft/in = 160 ft ✓

Area: Drawing: 4 × 6 = 24 in² Actual: 32 × 48 = 1,536 ft² Notice: 24 × 8² = 24 × 64 = 1,536 ft² ✓

MAPS:

Map scale examples: - 1:100,000 → 1 cm on map = 100,000 cm = 1 km in reality - 1 inch = 50 miles → Cities 2.5 inches apart are actually 125 miles apart

SOLVING SCALE PROBLEMS - STEPS:

1. Identify what you know:
   - Drawing size? Actual size?
   - Scale given?

2. Set up proportion:
   Drawing₁/Actual₁ = Drawing₂/Actual₂

3. Cross-multiply and solve

4. Check units: Convert if necessary (inches to feet, cm to m, etc.)

5. Verify: Does the answer make sense?

REAL-WORLD APPLICATIONS: - Architecture: Building blueprints - Urban planning: City maps - Engineering: Machine part diagrams - Model making: Trains, airplanes, cars - Geography: World maps, atlases - Interior design: Room layouts - Science: Cell diagrams, molecule models

Solve real-world scale drawing problems!

BASIC SCALE PROBLEMS: 1. A map scale is 1 cm = 25 km. If two cities are 4 cm apart on the map, what is the actual distance?

2. On a blueprint, 1 inch represents 12 feet. A room is drawn 3 inches wide. What is the actual width?

3. A model car has scale 1:24. If the model is 8 inches long, how long is the actual car?

4. A map has scale 1:50,000. A lake is 6 cm long on the map. What is its actual length in meters?
   (Hint: 1:50,000 means 1 cm = 50,000 cm = 500 m)

FINDING DRAWING SIZE: 5. A 90-foot building is drawn on a blueprint with scale 1 in = 15 ft. How tall is the building in the drawing?

6. A garden is 20 meters long. On a plan with scale 1 cm = 4 m, how long is the garden in the drawing?

7. A car is 18 feet long. In a model with scale 1 in = 3 ft, how long is the model?

FINDING THE SCALE: 8. A 40-foot room is represented by 2 inches on a drawing. What is the scale?

9. A 150-mile distance is shown as 3 cm on a map. What is the scale in cm per 50 miles?

AREA PROBLEMS:
10. A rectangular room on a blueprint is 2 in × 3 in. The scale is 1 in = 8 ft. Find:
    a) Actual dimensions
    b) Actual area
    c) Drawing area
    d) Verify: Drawing area × (scale factor)² = Actual area

11. A scale drawing of a park has scale 1 cm = 20 m. If the park's area in the drawing is 12 cm², what is the actual area in m²?

PERIMETER PROBLEMS:
12. A rectangular pool on a drawing is 5 cm × 10 cm with scale 1 cm = 2 m.
    a) What are the actual dimensions?
    b) What is the actual perimeter?

MULTI-STEP PROBLEMS: 13. A map has scale 1 inch = 40 miles. You measure 2.5 inches between City A and City B, and 3.2 inches between City B and City C. What is the total distance from A to C through B?

14. A rectangular garden is 30 ft × 45 ft. You want to draw it on paper where the length will be 9 inches.
    a) What scale should you use?
    b) How wide will the drawing be?

15. An architect's scale model of a building is 1:200. If the model is 60 cm tall, how tall is the actual building in meters?

CHALLENGE: 16. A square on a scale drawing has area 16 cm². The scale is 1 cm = 5 m. What is the actual area of the square in m²?

17. Two maps show the same city. Map A has scale 1:50,000 and Map B has scale 1:100,000. If a distance is 4 cm on Map A, what is it on Map B?

18. A rectangular room is 240 square feet in reality. On a drawing with scale 1 in = 8 ft, it's drawn as a 3 in × 2.5 in rectangle. Verify this is correct.

CRITICAL THINKING: 19. Why can't you just add the scale factor to dimensions? Explain why you must multiply.

20. If a scale is 1:100, by what factor do areas change? By what factor do volumes change?