United States
Hong KongScale drawings and similar figures
The rule names scaled drawings and models explicitly as examples under the ratio and proportion skill.
- Similar figures have corresponding sides in proportion.
- The scaling rule is the tested part: lengths scale by the factor, areas by its square, volumes by its cube.
On this page8 sections
Skill 2 of Competency 2 gives two examples in brackets: scaled drawings, and models. That is the rule telling you what a proportion question here looks like.
Similar figures
Two figures are similar when they have the same shape and different size. Corresponding angles are equal and corresponding sides are in a constant ratio, and that ratio is the scale factor.
Everything else follows from those two sentences. If a triangle with sides 3 and 5 is similar to one with a corresponding side of 6, the scale factor is 2 and the other side is 10.
The setup
- Identify which sides correspond, usually by position or by the order of the letters naming the figures
- Write a fraction of two corresponding sides, small over large in both fractions
- Cross multiply and solve
Step two carries the error. Mixing the direction, putting the small figure on top in one fraction and the large one on top in the other, gives an answer that is wrong by a factor equal to the square of the scale. Keep the labels.
The scaling rule
| Quantity | Scales by |
|---|---|
| Length | The scale factor |
| Area | The factor squared |
| Volume | The factor cubed |
Say it in words rather than numbers, because the words are what you will remember under pressure. Lengths scale by the factor. Areas scale by the factor squared. Volumes scale by the factor cubed.
This is counterintuitive and it is exactly why it makes a good item. Double the sides of a room and the carpet needed goes up four times, not two, and a candidate answering from intuition will find their intuitive answer sitting there as option B.
A model of a building is built to a scale of 1 to 10. The model has a floor area of 2 square centimeters. What is the floor area of the real building, in square centimeters?
- 20
- 40
- 100
- 200
Map and drawing questions
These give you a scale in the form "1 centimeter represents 50 kilometers" and a measurement on the drawing. Set it up as a proportion with the drawing units on top in both fractions and the real units underneath, then cross multiply.
Read the scale carefully. A scale written as a ratio with no units, such as 1 to 50, means the same unit on both sides, so a centimeter on the model is 50 centimeters in reality. A scale written with units may mix them, and the units are the whole instruction.
Why this is worth more than one question of preparation
Because the scaling rule shows up in three places: similar figures, scale models, and any measurement item that changes a dimension. It is one idea that covers several skills across a competency we estimate at about seven of the 35 questions, and that estimate is our arithmetic on the rule's skill counts rather than a published weighting.
It is also the piece of school mathematics that most adults have genuinely forgotten rather than merely rusted on. Worth twenty minutes with a pencil and a drawing.
Use one scale factor for corresponding lengths
A scale drawing enlarges or reduces an object while preserving proportional dimensions. If 1 centimeter represents 4 meters, a drawing length of 3.5 centimeters represents 14 meters. The scale factor compares corresponding lengths; use the same direction consistently. If the scale is drawing-to-real, multiply drawing measurements by the real-units-per-drawing-unit factor.
Similar figures have equal corresponding angles and proportional corresponding side lengths. If a 5-centimeter side in a small figure corresponds to 8 centimeters in a larger figure, the scale factor from small to large is 8/5 = 1.6. A 3-centimeter side in the small figure corresponds to 4.8 centimeters in the larger one. Pair corresponding sides in the same order.
Keep units compatible
Convert units before using the scale. If a map scale is 1 inch = 20 miles and a route measures 2.75 inches, the real distance is 55 miles. If the route is given in centimeters, first convert to inches or express the scale in centimeters. Mixing measurement units inside a proportion creates a numerical result with no useful meaning.
A proportion can solve an unknown: drawing length / real length = drawing scale ratio. If 2 inches represents 30 miles, then 5 inches represents x miles: 2/30 = 5/x, so 2x = 150 and x = 75. Check direction: the longer drawing segment should represent the longer real distance.
Understand how area and volume scale
Lengths scale by a factor k. Areas of similar figures scale by k², and volumes of similar solids scale by k³. If each side of a model is doubled, its surface area becomes four times as large and its volume becomes eight times as large. This is why a linear scale factor cannot be applied directly to area or volume.
For example, a square with side 3 centimeters has area 9 square centimeters. An enlarged similar square with scale factor 2 has side 6 centimeters and area 36 square centimeters, four times the original. For a rectangular prism, scale all three dimensions and multiply to see the cubic effect. Keep units squared or cubed in the final answer.
- Identify the scale direction and corresponding measurements.
- Pair corresponding sides in the same order.
- Convert units before setting up a proportion.
- Check whether the question asks for length, area, or volume.
- Apply k to lengths, k² to areas, and k³ to volumes.
Common questions
What makes two figures similar?
Equal corresponding angles and corresponding sides in a constant ratio. That ratio is the scale factor, and everything else follows from it: knowing one pair of corresponding sides gives you the factor, and the factor gives you every other side.
If I double the sides, what happens to the area?
It goes up four times, because areas scale by the square of the length factor. Volumes scale by the cube, so doubling the sides multiplies volume by eight. The intuitive answer of doubling is wrong and is usually offered as an option.
How do I read a scale on a drawing?
A scale written as a plain ratio, such as 1 to 50, uses the same unit on both sides. A scale written with units, such as one centimeter representing 50 kilometers, mixes them, and the units are the instruction. Set both fractions up with the same unit on top.
Are scale drawings actually on the FTCE?
The rule names them. Skill 2 of Competency 2 reads "Solve problems involving ratio and proportion (e.g., scaled drawings, models, real-world scenarios)", so scaled drawings and models are the state’s own examples of what this skill covers.