Scale Drawings and Similar Figures
Use scale factors to find actual measurements from drawings and models. Identify similar figures by equal ratios of corresponding sides, and solve for missing lengths using proportions.
A lesson of Pre-Algebra, in Ratios, Rates & Proportional Reasoning, about 10 minutes.
By the end of this lesson, you will be able to use scale factors to find actual and drawing measurements and identify similar figures using proportional sides.
Scale Drawings
A scale drawing represents a real object with all dimensions reduced (or enlarged) by the same ratio — the scale factor.
Scale: "1 inch = 20 feet" means 1 inch on the drawing represents 20 feet in real life.
Finding actual length: $$\text{Actual} = \text{drawing measurement} \times \text{scale factor}$$
Example: On a map with scale 1 cm = 50 km, two cities are 4.5 cm apart.
Actual distance = $4.5 \times 50 = 225$ km
Finding drawing length: $$\text{Drawing} = \text{actual measurement} \div \text{scale factor}$$
Similar Figures
Similar figures have the same shape but different sizes. Their corresponding angles are equal and their corresponding sides are proportional.
Finding a missing side: set up a proportion between corresponding sides.
Example: Two similar triangles. One has sides 3, 4, 5. The other has its smallest side = 9. Find the other sides.
Scale factor = $9 \div 3 = 3$
Other sides: $4 \times 3 = 12$ and $5 \times 3 = 15$
Perimeter and Area of Similar Figures
If the scale factor between two similar figures is $k$:
- Perimeters are in ratio $k$
- Areas are in ratio $k^2$
Example: Two similar rectangles; one has dimensions 4×6, the other 8×12 (scale factor = 2).
Perimeters: 20 and 40 → ratio 2 ✓
Areas: 24 and 96 → ratio 4 = $2^2$ ✓
Summary
Scale factor: multiply drawing by scale to get actual; divide actual by scale to get drawing.
Similar figures: same shape, proportional sides, equal angles.
Missing sides: set up proportions between corresponding parts.
Areas scale by $k^2$ when lengths scale by $k$.