Scale Diagrams and Similar Shapes

Same shape, different size — every side scales by the exact same amount.

🧱 See It

A scale diagram is a smaller or larger copy of a real object where every single measurement scales by the same factor — a map, a blueprint, a model car. Two shapes are similar when they have the same shape but possibly a different size: corresponding angles are always exactly equal, and corresponding sides are always exactly proportional — never just "close."

✏️ Draw It

4 3 5

Triangle A

8 6 10

Triangle B (scale factor 2)

Every side of Triangle B is exactly double the matching side of Triangle A: 3→6, 4→8, 5→10. Same angles, same shape, just bigger.

🔢 Write It

scale factor = new length ÷ original length

Once the scale factor is known from one pair of corresponding sides, every other side scales by that exact same factor. Use cross-multiplication (from Proportional Reasoning) to find any missing side.

Corresponding means "matching position" — the shortest side of one shape matches the shortest side of the other, the longest matches the longest, and so on.

💡 Worked Examples

Example 1 — A scale drawing has a scale factor of 1:50 (1 cm on paper = 50 cm real life). A wall measures 6 cm on the drawing. How long is the real wall?

  1. Set up the scale: 1 cm / 50 cm = 6 cm / x cm
  2. Cross-multiply: 1 × x = 50 × 6
  3. Simplify: x = 300 cm
  4. Convert to a friendlier unit: 300 cm = 3 m
  5. Answer: 3 m

Example 2 — Triangle A has sides 3, 4, 5. Triangle B is similar, with its shortest side = 6. Find Triangle B's other two sides.

  1. Find the scale factor from the known pair: 6 ÷ 3 = 2
  2. Apply that same factor to the middle side: 4 × 2 = 8
  3. Apply it to the longest side too: 5 × 2 = 10
  4. Answer: 8 and 10

🧠 Quick Reference

Similar shapes: same shape, equal corresponding angles, all corresponding sides scale by the SAME factor. Find the scale factor from one known pair of sides first, then apply that exact factor to every other side.
👪 For Parents & Tutors

BC Curriculum (Level 4 / Grade 9, Geometry): "Spatial proportional reasoning: scale diagrams, similar triangles and polygons, linear unit conversions."

Watch for: assuming two shapes are similar just because they "look similar," without actually checking that the angles match and the sides are truly proportional (not just similar-looking). Always verify the scale factor works for EVERY pair of corresponding sides, not just the one that was easiest to spot.

Comes up again: this page directly depends on Level 3's Proportional Reasoning, and connects closely to the Pythagorean Theorem — similar right triangles show up together often (as in the 3-4-5 / 6-8-10 pair above).