Scale Drawings - Seventh Grade
Scale Factor, Proportions & Dimensional Changes
1. Understanding Scale Drawings
What is a Scale Drawing?
A scale drawing is a representation of an object
where all dimensions are PROPORTIONALLY
increased or decreased
Examples: Maps, blueprints, architectural plans, models
What is Scale?
Scale is a RATIO that compares:
• Drawing dimension : Actual dimension
• Or Model size : Real size
Common notation: 1 cm = 5 m or 1:500
2. Scale Factor
Definition
Scale Factor (k) is the multiplier
that relates the size of the drawing
to the size of the actual object
Scale Factor Formula
k = New Dimension / Original Dimension
or
k = Drawing Length / Actual Length
If k > 1: The drawing is LARGER than the actual object (enlargement)
If k < 1: The drawing is SMALLER than the actual object (reduction)
If k = 1: The drawing is the SAME SIZE as the actual object
Using Scale Factor
Actual Length = Drawing Length × k
or
Drawing Length = Actual Length ÷ k
3. Solving Scale Drawing Problems
Method: Setting Up Proportions
Drawing₁ / Actual₁ = Drawing₂ / Actual₂
Set up a proportion and solve by cross-multiplying
Example 1: Finding Actual Length
Problem: A room measures 6 inches long on a blueprint with a scale of 1 inch = 4 feet. What is the actual length?
Method 1: Using Proportion
1 inch / 4 feet = 6 inches / x feet
1 × x = 4 × 6
x = 24 feet
Method 2: Using Scale Factor
Scale factor k = 4 feet / 1 inch = 4
Actual length = 6 × 4 = 24 feet
Answer: The actual length is 24 feet
Example 2: Finding Drawing Length
Problem: A building is 60 feet tall. On a scale drawing where 1 cm = 10 feet, how tall is the building in the drawing?
Solution:
1 cm / 10 feet = x cm / 60 feet
10 × x = 1 × 60
10x = 60
x = 6 cm
Answer: The building is 6 cm tall in the drawing
Example 3: Finding Scale Factor
Problem: A model car is 5 inches long. The actual car is 15 feet long. What is the scale factor?
Step 1: Convert to same units
15 feet = 15 × 12 = 180 inches
Step 2: Find scale factor
k = Drawing / Actual = 5 / 180 = 1/36
Answer: Scale factor is 1:36 or 1 inch = 3 feet
4. Perimeter Changes with Scale
The Rule
New Perimeter = k × Original Perimeter
Where k = scale factor
Perimeter changes by the SAME factor as length
Key Point:
If you double the dimensions (k = 2),
the perimeter also DOUBLES
Example
Problem: A rectangle has dimensions 4 cm by 6 cm. If we scale it by a factor of 3, what is the new perimeter?
Step 1: Find original perimeter
P = 2(4 + 6) = 2(10) = 20 cm
Step 2: Apply scale factor
New perimeter = 3 × 20 = 60 cm
Verification:
New dimensions: 12 cm by 18 cm
New perimeter = 2(12 + 18) = 60 cm ✓
Answer: New perimeter = 60 cm
5. Area Changes with Scale
The Rule
New Area = k² × Original Area
Where k = scale factor
Area changes by the SQUARE of the scale factor
Key Point:
If you double the dimensions (k = 2),
the area becomes 4 TIMES larger (2² = 4)
If you triple (k = 3), area becomes 9 times larger (3² = 9)
Why k² for Area?
Area = length × width
When scaled: New Area = (k × length) × (k × width)
New Area = k² × (length × width) = k² × Original Area
Example
Problem: A rectangle has area 24 cm². If we scale it by a factor of 3, what is the new area?
Solution:
New area = k² × Original area
New area = 3² × 24
New area = 9 × 24
New area = 216 cm²
Answer: New area = 216 cm²
Complete Example
Problem: A square has side 5 cm. Scale factor k = 4. Find new perimeter and area.
Original measurements:
Side = 5 cm
Perimeter = 4 × 5 = 20 cm
Area = 5² = 25 cm²
New measurements (k = 4):
New side = 4 × 5 = 20 cm
New perimeter = 4 × 20 = 80 cm (or 4 × original)
New area = 4² × 25 = 16 × 25 = 400 cm²
Perimeter increased by 4×, Area increased by 16×!
6. Common Scale Notations
| Notation | Meaning | Example |
|---|---|---|
| 1 cm = 5 m | 1 cm in drawing represents 5 m in reality | Maps, blueprints |
| 1:100 | 1 unit in drawing = 100 units actual | Architectural plans |
| 1 in : 4 ft | 1 inch in drawing = 4 feet actual | House plans |
| Scale 1/50 | Drawing is 1/50 the size of actual | Models |
Important: Always make sure units match when calculating!
Quick Reference: Scale Factor Effects
| Scale Factor (k) | Length Change | Perimeter Change | Area Change |
|---|---|---|---|
| k = 2 | ×2 | ×2 | ×4 (2²) |
| k = 3 | ×3 | ×3 | ×9 (3²) |
| k = 4 | ×4 | ×4 | ×16 (4²) |
| k = 1/2 | ÷2 | ÷2 | ÷4 (1/4) |
| k = 5 | ×5 | ×5 | ×25 (5²) |
💡 Important Tips to Remember
✓ Scale drawing: Proportional representation of actual object
✓ Scale factor k: Drawing dimension / Actual dimension
✓ Finding actual: Multiply drawing by k
✓ Finding drawing: Divide actual by k
✓ Units must match: Convert before calculating!
✓ Perimeter changes: Multiply by k (linear)
✓ Area changes: Multiply by k² (quadratic)
✓ Double size (k=2): Perimeter ×2, Area ×4
✓ Triple size (k=3): Perimeter ×3, Area ×9
✓ Proportions work: Set up ratios and cross-multiply
🧠 Memory Tricks & Strategies
Scale Factor:
"New over old, that's how it's told - scale factor makes shapes bold!"
Perimeter vs Area:
"Perimeter's k, that's the key - but area's k squared, you'll see!"
Why k² for Area:
"Two dimensions multiply, so scale factor gets squared - that's why!"
Finding Actual Size:
"Drawing times scale makes it real, that's the measurement deal!"
Units:
"Before you calculate, make units mate - or your answer won't be great!"
Scale of 1:100:
"One to a hundred means quite small - multiply by 100 to get it all!"
Master Scale Drawings! 📐 🗺️
Remember: Perimeter ×k, Area ×k²!