Scale and similarity

Similar shapes are the same shape at different sizes: a photo and its enlargement, a model car and the real one, a plan and the house it describes. In this topic you'll use similar right-angled triangles to find lengths you can't measure directly, and work out what an enlargement does to lengths, areas and volumes. This is the groundwork for trigonometry in the next topic. Diagrams are not drawn to scale unless the question asks you to measure them.

What you'll learn

• Investigation: algorithms in geometry. Follow, test and improve step-by-step methods for cutting a line into equal parts, deciding whether triangles are similar, and drawing enlargements.
• 7.01 Similar right-angled triangles: why one matching acute angle is enough, naming the hypotenuse, opposite and adjacent sides, finding missing sides, and why side ratios stay the same for a given angle.
• 7.02 Scale factors: enlargements and reductions, centres of enlargement, and scales such as 1 : 100 on plans, models and maps.
• Investigation: guessing distance. Estimate heights and distances outside with the shadow method, the mirror method and the thumb jump, then judge how good each model is.
• 7.03 Scale and area: when lengths are multiplied by $k$, areas are multiplied by $k^2$. Areas of similar shapes, and real areas from plans and maps.
• 7.04 Scale and volume: when lengths are multiplied by $k$, volumes are multiplied by $k^3$. Volumes, capacities and masses of similar solids.
• Investigation: scale diagrams. Read a plan printed at true size, solve problems by drawing accurate scale diagrams, and find out how accurate they are.

Which level should I pick?

Every worksheet comes in three levels. Each one covers the same skills.

• Level 1 · Warm-up: whole-number scale factors, diagrams with the matching parts marked, and more help along the way. A good place to start.
• Level 2 · Core: the main worksheet for the lesson.
• Level 3 · Challenge: working backwards, algebra, overlapping triangles and problems with more steps and more explaining, for when the Core level feels easy.

Not sure? Start with Level 2. If the first few questions feel tricky, drop to Level 1. If you finish quickly and get them right, try Level 3. Download the worksheet, print it, and write your answers in the boxes.

Practice tests

When you've finished the lessons, try the three practice tests. Each one is like the real topic test: 45 minutes, 40 marks, a section without a calculator and a section with one. Write your working under each question, then copy your final answers onto the answer sheet on the last page, just like you will in the real test.