Mathematics II · Calculator allowed

Scale Factor and Similar Shapes

Two shapes can be different sizes but the exact same shape. When they are, one simple number lets you find any missing side.

Think of a photo and a thumbnail of that photo. Same picture, just smaller. Nothing is stretched or squished, it is shrunk by the same amount in every direction. In math, shapes like that are called similar, and the amount you shrink or grow by is the scale factor.

This shows up all over the calculator part of the CAEC: maps, blueprints, scale models, and "find the missing length" problems. Once you can spot similar shapes and set up a simple proportion, these become some of the most reliable points on the test. Let's build it up step by step.

What makes two shapes "similar"

Similar is a precise math word, not just "kind of alike." Two shapes are similar when both of these are true:

Equal angles. Every corner matches its partner exactly. A right angle stays a right angle; the shape is never tilted or bent.
Proportional sides. Each side of the bigger shape is the same number of times longer than the matching side of the smaller shape. That shared number is the scale factor.

Memorize this one: scale factor is not on the CAEC formula sheet. The sheet gives you area and volume formulas, but the idea of matching up corresponding sides and finding a scale factor is something you need to know on your own. The good news: it is just a ratio.

A picture of similar triangles

Here are two similar triangles. The big one is exactly twice the size of the small one, so every matching side is doubled. Notice the angles are identical, only the size changed.

3 → 6, 4 → 8, 5 → 10. Every side is multiplied by 2, so the scale factor from small to large is 2.

Two ways to find a missing side

There are two reliable methods, and they always agree. Use whichever feels clearer to you on a given problem.

Method 1, Scale factor. Find the scale factor by dividing one matching pair of sides (new ÷ old). Then multiply (or divide) the known side by that factor to get the missing one.
Method 2, Proportion. Set up two equal ratios of matching sides and cross-multiply to solve. This is great when the numbers are not clean.

The golden rule: always match corresponding sides, the ones in the same position on each shape. Long side goes with long side, short with short. Mixing them up is the number-one mistake.

Worked example #1: find the missing side with a scale factor

Two similar rectangles. The small one is 4 cm by 6 cm. The large one is 12 cm wide on the side that matches the 4 cm side. How tall is the large rectangle?

Scale factor: match the known pair, 4 cm and 12 cm. 12 ÷ 4 = 3. So the large shape is 3 times the small one.
Apply it: the other small side is 6 cm, so the matching large side is 6 × 3 = 18 cm.

scale factor = new ÷ old
             = 12 ÷ 4
             = 3

missing side = 6 × 3
             = 18 cm

Answer: 18 cm. Quick sanity check, the large shape should be bigger, and 18 > 6, so the answer makes sense.

Worked example #2: find the missing side with a proportion

Two similar triangles. The small triangle has a base of 5 and a height of 8. The large triangle's matching base is 15. Find its height, h. The numbers are not a clean multiple this time, so a proportion keeps us tidy.

Set up matching ratios: small base over large base equals small height over large height.
Cross-multiply: multiply across the equals sign, then solve for h.

515 = 8h(base/base = height/height)

5 × h = 15 × 8 (cross-multiply)

5h = 120

h = 120 ÷ 5 = 24

Answer: h = 24. Check with the scale factor: 15 ÷ 5 = 3, and 8 × 3 = 24. Both methods agree, which is exactly what should happen.

Worked example #3: a map scale

Maps and blueprints are just scale drawings. A map says 1 cm represents 25 km. Two cities are 7 cm apart on the map. How far apart are they in real life?

Read the scale: every 1 cm on the map stands for 25 km on the ground.
Multiply: 7 cm means 7 groups of 25 km, so 7 × 25 = 175 km.

scale: 1 cm  →  25 km

real distance = 7 × 25
              = 175 km

Answer: 175 km. To go the other way (real distance back to map distance), you would divide instead: 175 ÷ 25 = 7 cm.

Tips that keep you from slipping

Match corresponding sides first. Before any math, pair up the sides that sit in the same spot on each shape. Long with long, short with short.
Decide which way you are scaling. Going from small to large, you multiply by the scale factor. Going from large to small, you divide. A quick "should this get bigger or smaller?" check catches most errors.
Keep your units the same. With maps and scale drawings, make sure both sides of the proportion use the same units (all cm, or all km) before you solve.
Use your calculator, but estimate too. This is the calculator section, so let the calculator do the division. Still, a rough estimate tells you instantly if you typed something wrong.

Your turn: practice problems

Match the sides, find the scale factor or set up a proportion, then solve. No peeking until you have tried.

Two similar rectangles. The small one is 3 cm by 7 cm. The large one's matching short side is 9 cm. Find its long side.
Two similar triangles. Small base 4, large matching base 10. Small height 6, find the large height.
A scale model uses 2 cm to represent 5 m. A wall on the model is 14 cm long. How long is the real wall?
Find the scale factor from a small shape with an 8 cm side to a large shape whose matching side is 20 cm.

Tap to reveal the answers

1. Scale factor = 9 ÷ 3 = 3. Long side = 7 × 3 = 21 cm.
2. Scale factor = 10 ÷ 4 = 2.5. Large height = 6 × 2.5 = 15. (Or proportion: 4⁄10 = 6⁄h, so 4h = 60, h = 15.)
3. 2 cm stands for 5 m, so 1 cm stands for 2.5 m. Real wall = 14 × 2.5 = 35 m.
4. Scale factor = new ÷ old = 20 ÷ 8 = 2.5.

Why this matters for the CAEC

Scale factor and similar shapes power a whole family of CAEC questions: reading maps, working with blueprints and models, and finding missing lengths in geometry figures. Because scale factor is not on the formula sheet, the test is really checking that you understand the idea, matching sides and a single ratio. Get comfortable with that and these become quick, dependable points.

Want more practice like this? Our CAEC math guide and the CAEC Ready Workbook are packed with worked examples and practice questions, or start with a free math sample to test yourself.

Disclaimer

This article is a general math tutorial for study purposes. CAEC Ready is an independent study resource and is not affiliated with or endorsed by any government, ministry of education, or official CAEC testing provider.