Similarity of shapes · Code with Martin

The same shape, a different size

A photo on a phone and the same photo on a poster. A floor plan of a flat and the real flat. A model car and a real car. Each time the same shape, a different size. Maths calls that similarity.

Two shapes are similar when one comes from the other by even enlarging or shrinking. Even — that is the key word. All lengths are multiplied by the same number.

That number is called the similarity coefficient and it is written k. When k = 3, every side of the large shape is three times longer than the matching side of the small one. When k = 0.5, everything has shrunk to a half.

And the angles? Angles do not change at all. An enlarged photo does not have more slanted corners. That is why similar shapes look “the same”, even if one fits in a palm and the other on a billboard.

Today you will learn to find the coefficient, use it to calculate sides and spot when shapes are not similar.

Sofie is printing the school logo on two things: a business card and a T-shirt. On the card the logo is 4 cm wide and 3 cm high. On the T-shirt she wants it 20 cm wide. “How high should I set it?” she wonders. Jonáš counts: “The width grew from 4 to 20 — that is five times. So the height must be five times too: 3 times 5 is 15 cm.” Sofie sets 20 × 15 and the logo looks alive. “If I just guessed the height, a pancake or a noodle would come out. The coefficient guards the shape.”

💡

You always find the coefficient the same way: k = side of the large / matching side of the small. Watch that you divide matching sides — longest with longest, shortest with shortest.

The similarity coefficient step by step

Step 1: Pair matching sides. In similar shapes, sides match by position: the longest belongs with the longest, the shortest with the shortest.

Step 2: Calculate the coefficient. k = length of a side of the large / length of the matching side of the small. Triangles with sides 2, 3, 4 and 6, 9, 12: k = 6/2 = 3. Check on the other sides: 9/3 = 3 and 12/4 = 3. The same number everywhere — they are similar.

Step 3: Use the coefficient on an unknown side. Side of the large = side of the small · k. Side of the small = side of the large / k. Nothing more.

When similarity is NOT there: triangles 2, 3, 4 and 4, 6, 10. Ratios: 4/2 = 2, 6/3 = 2, but 10/4 = 2.5. One number ducks away — they are not similar. The last side stretched more than the others and the shape warped.

Watch the area! When k = 3, the sides are three times longer, but the area is nine times bigger (3² = 9). Enlarging grows in both directions at once — in width and in height. A photo enlarged three times needs nine times more paper. This is a favourite trick question, so write it in bold.

Example 1: are they similar?

Question: Triangle A has sides 3, 5, 6 cm. Triangle B has sides 9, 15, 18 cm. Are they similar?

Method: I pair sides by size and divide: 9/3 = 3, 15/5 = 3, 18/6 = 3.

All the ratios are the same.

Answer: Yes, they are similar with coefficient k = 3. Each side of triangle B is three times the matching side of A. If a single ratio came out different, the answer would be no — you check all the sides, not just one.

Example 2: find a missing side

Question: A rectangular poster is similar to a postcard with k = 4. The postcard has size 10 × 15 cm. What size is the poster? And how many times more paper does it use?

Method: I multiply the sides by the coefficient: 10 · 4 = 40 cm and 15 · 4 = 60 cm.

Areas: postcard 10 · 15 = 150 cm², poster 40 · 60 = 2400 cm².

Ratio of areas: 2400/150 = 16. And 16 = 4².

Answer: The poster is 40 × 60 cm and uses sixteen times more paper — area grows with the square of the coefficient.

Example 3: shrinking, k smaller than 1

Question: A pitch has size 40 × 20 m. You draw it on a plan shrunk with coefficient k = 0.01 (so a hundred times smaller). What size will it be on the plan?

Method: I multiply by the coefficient: 40 · 0.01 = 0.4 m = 40 cm. And 20 · 0.01 = 0.2 m = 20 cm.

Answer: On the plan the pitch will be 40 × 20 cm.

A coefficient smaller than 1 means shrinking — it is the same as a map scale of 1 : 100. Similarity and scale are the same idea in two coats.

⚠️

Area does not grow with k, but with k². Triple the sides = nine times the area. A classic entrance-exam trap says: “the sides grew three times, how many times did the area grow?” Who answers three times loses a mark. Multiply lengths by k, areas by k², volumes even by k³.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

Training on the coefficient: side of the small times k = side of the large. Write a whole number.

Practise the questions

On paper

Coefficients, missing sides and one trap on area.

Triangles 4, 6, 8 cm and 10, 15, 20 cm: check similarity with all three ratios and write k.
A photo 9 × 13 cm is enlarged with k = 3. What will the size be? And how many times does the area grow? Calculate both areas and check that the ratio is k².
Rectangles 6 × 10 cm and 9 × 16 cm: are they similar? Decide with the ratios and give a reason.
A flat plan has scale 1 : 50. A room is 8 cm long on the plan. How long is it in real life? (Scale = the coefficient.)

Now you 💪

You calculate the coefficient as a ratio of matching sides and check it on all sides.
You can find a missing side both ways: times k when enlarging, divided by k when shrinking.
You know that area grows with k², and you can show it on concrete numbers.

✅

Done when: You have checked similarity through all the ratios, found missing sides, spotted a non-similar pair and calculated the area jump k².

What to take from this lesson

Similar shapes have the same shape: angles the same, lengths multiplied by the coefficient k.
k = side of the large / matching side of the small. Check on all sides.
k bigger than 1 enlarges, k smaller than 1 shrinks. A map scale is k too.
Area grows with k², volume with k³ — not with k.
One ratio that ducks away = the shapes are not similar.