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SCALE

Scale Drawing
Problems

Master scale factors, compute actual lengths and areas from drawings, and reproduce drawings at different scales.

7th Grade Math

What Is a Scale Drawing?

A scale drawing is a proportional representation of a real object. Every length in the drawing is multiplied by the same constant ratio compared to the actual object.

Maps
Roads and distances shrunk to fit on paper
Blueprints
Buildings and rooms drawn at a fraction of real size
Models
Toy cars, globes, and dioramas that preserve shape

Understanding the Scale Factor

A scale is written as a ratio. For example, \(1 : 50\) means every \(1\) unit on the drawing represents \(50\) units in real life.

Drawing
\(2 \text{ cm}\)
\(\times 50\)
 Actual
\(100 \text{ cm} \, (1 \text{ m})\)

Scale Factor as a Fraction

The scale factor \(k\) tells you how many times bigger or smaller the drawing is compared to the real object:

\(k < 1\)
Drawing is smaller than reality (maps, floor plans)
\(k = 1\)
Drawing is actual size (full-scale)
\(k > 1\)
Drawing is larger than reality (microscope diagrams)

Core Rule: Scaling Lengths

Equivalently, if the scale is \(1 : n\), then:

This works for every linear measurement: width, height, perimeter, diagonal, circumference.

Core Rule: Scaling Areas

Area involves two dimensions, so the scale factor is squared:

Drawing
\(3 \text{ cm} \times 3 \text{ cm}\)
\(= 9 \text{ cm}^2\)
\(\text{Scale } 1:4\)
 Actual
\(12 \text{ cm} \times 12 \text{ cm} = 144 \text{ cm}^2\)
\(\text{Lengths: } \times 4\)
\(\text{Area: } \times 16\)

Length vs. Area Scaling

Length Area
Formula \(\text{Actual} = \text{Drawing} \times n\) \(\text{Actual} = \text{Drawing} \times n^2\)
Scale \(1 : 5\) Multiply by 5 Multiply by 25
Scale \(1 : 10\) Multiply by 10 Multiply by 100
Scale \(1 : 100\) Multiply by 100 Multiply by 10{,}000

How to Solve Scale Problems

1
Identify the scale. Write it as a ratio \(1 : n\) or as a fraction \(k\).
2
Read the given measurement from the drawing (in drawing units).
3
Decide: length or area? Multiply by \(n\) for length, or by \(n^2\) for area.
4
Convert units if necessary (cm to m, inches to feet, etc.).
5
State your answer with correct units.

Example 1: Actual Length

A floor plan uses a scale of \(1 : 25\). A room measures \(6\) cm long on the drawing. What is the actual length?

Drawing
\(6 \text{ cm}\)
 →
\(\times 25\)
 Actual
\(150 \text{ cm} = 1.5 \text{ m}\)

Example 2: Actual Area

On the same floor plan (scale \(1 : 25\)), a room measures \(6\) cm by \(4\) cm. What is the actual area?

Drawing
\(6 \text{ cm}\)
\(4 \text{ cm}\)
\(24 \text{ cm}^2\)
→
\(\times 25^2 = \times 625\)
Actual
\(150 \text{ cm} \times 100 \text{ cm}\)
\(= 15{,}000 \text{ cm}^2 = 1.5 \text{ m}^2\)

Example 3: New Scale

A map uses scale \(1 : 500\). A park is \(8\) cm wide on this map. You need to reproduce it at scale \(1 : 200\). How wide is it on the new map?

1
Find actual width: \(8 \text{ cm} \times 500 = 4{,}000 \text{ cm}\)
2
Apply new scale: \(\displaystyle\frac{4{,}000}{200} = 20 \text{ cm}\)
3
Shortcut: Combine in one step: \(\displaystyle 8 \times \frac{500}{200} = 8 \times 2.5 = 20 \text{ cm}\)

Example 4: Scaling a Triangle

A triangle on a scale drawing (scale \(1 : 10\)) has sides \(3\) cm, \(4\) cm, and \(5\) cm. Find the actual perimeter and area.

\(4 \text{ cm}\)
\(3 \text{ cm}\)
\(5 \text{ cm}\)
 Drawing (1:10) →
\(\times 10\)
\(40 \text{ cm}\)
\(30 \text{ cm}\)
\(50 \text{ cm}\)
 Actual

Example 5: Finding the Scale

A building is \(45\) m tall in real life. On a drawing, it is \(9\) cm tall. What is the scale of the drawing?

1
Convert to same units: \(45 \text{ m} = 4{,}500 \text{ cm}\)
2
Write the ratio: \(\displaystyle\frac{\text{Drawing}}{\text{Actual}} = \frac{9}{4{,}500}\)
3
Simplify: \(\displaystyle\frac{9}{4{,}500} = \frac{1}{500}\), so the scale is \(1 : 500\)

Common Mistakes

✕ Using \(n\) for area
Area scales by \(n^2\), not \(n\). If scale is \(1:5\), area multiplies by 25, not 5.
✕ Mismatched units
Comparing cm on the drawing to m in reality without converting first leads to wrong ratios.
✕ Dividing instead of multiplying
To find actual from drawing, multiply by \(n\). To find drawing from actual, divide by \(n\).

Summary of Scale Formulas

Task Formula Example (\(1:20\))
Drawing → Actual Length \(\text{D} \times n\) \(5 \text{ cm} \times 20 = 100 \text{ cm}\)
Actual → Drawing Length \(\text{A} \div n\) \(100 \text{ cm} \div 20 = 5 \text{ cm}\)
Drawing → Actual Area \(\text{D} \times n^2\) \(25 \text{ cm}^2 \times 400 = 10{,}000 \text{ cm}^2\)
Actual → Drawing Area \(\text{A} \div n^2\) \(10{,}000 \text{ cm}^2 \div 400 = 25 \text{ cm}^2\)
Change Scale \(\text{Old} \times \frac{n_{\text{old}}}{n_{\text{new}}}\) \(5 \text{ cm} \times \frac{20}{10} = 10 \text{ cm}\)

Question 1 of 5

A map has a scale of \(1 : 200\). A street measures \(7\) cm on the map. What is the actual length of the street?

Question 2 of 5

A scale drawing uses a scale of \(1 : 50\). A rectangular room on the drawing is \(4\) cm by \(3\) cm. What is the actual area of the room?

Question 3 of 5

A building is \(60\) m tall. On a drawing, it appears \(12\) cm tall. What is the scale of the drawing?

Question 4 of 5

A park is \(10\) cm wide on a map drawn at scale \(1 : 300\). You redraw it at scale \(1 : 150\). How wide is the park on the new drawing?

Question 5 of 5

A square garden has an actual area of \(400\) m². On a scale drawing (\(1 : 100\)), what is the side length of the garden on the drawing?