Similarity in practice

Measure what you cannot reach

You have the tests, you can do the coefficient. Today we put similarity to work for real — on things a tape measure cannot reach: the height of a tree, the width of a river, a distance on a map.

The idea stays the same. You find two similar triangles: one small one you can measure, and one large one that holds the unknown. The coefficient carries the measurement from the small to the large.

The most famous trick is the shadow method. The sun shines on everything at the same angle. You and your shadow make a triangle. A tree and its shadow make a similar triangle (test AA: a right angle at the ground + the same sun angle). You measure yourself, your shadow and the tree’s shadow — the height of the tree drops out.

The second big tool is a map. Scale 1 : 25 000 is a similarity coefficient: reality is 25 000 times bigger than the paper.

This is the lesson where geometry stops being about pictures in a notebook and starts measuring the world. Take a tape measure, we are going out — at least in our heads.

Filip bets Ema he will find the height of the pitch mast without climbing. He waits for sun, stands next to the mast and measures: “I am 1.5 metres and my shadow is 2 metres. The mast’s shadow… 16 metres.” Ema counts: “Your triangle: height 1.5, shadow 2. The mast: height ?, shadow 16. The shadow stretched eight times, so the height too: 1.5 times 8 is 12 metres.” Filip raises a fist: “Won. The sun measured for us.”

Three tools from similarity

Tool 1: The shadow method. Method: 1) measure your height and the length of your shadow, 2) measure the object’s shadow, 3) calculate the coefficient k = object’s shadow / your shadow, 4) height of the object = your height · k. It works for a tree, a mast, a house — anything upright on flat ground.

Tool 2: A map and scale. Scale 1 : 25 000 says: 1 cm on the map = 25 000 cm in reality = 250 m. Method: 1) measure the distance on the map with a ruler, 2) multiply by the scale, 3) convert units (cm → m → km). A distance of 6 cm on a map 1 : 25 000 is 6 · 25 000 = 150 000 cm = 1.5 km.

Tool 3: A triangle across a river. You measure the width of a river from one bank: on your bank you set out a small triangle similar to a large one (which reaches across the water) and convert with the coefficient. The same idea, just lying down.

A shared method for all three: sketch a picture, mark both triangles, write why they are similar (almost always AA), calculate k from the pair you know, and multiply. And at the end a sense check: a tree 12 m high is reasonable, a tree 480 m high is not. If nonsense comes out, you probably flipped the coefficient.

Example 1: the height of a tree from a shadow

Question: A stick 1.2 m long casts a shadow 0.8 m. A tree next to it casts a shadow 6 m. How high is the tree?

Method: Both triangles have a right angle at the ground and the same sun angle → similar by AA.

Coefficient from the shadows: k = 6/0.8 = 7.5.

Height of the tree = 1.2 · 7.5 = 9 m.

Sense check: a nine-metre tree is a normal grown tree. And the ratio matches: the tree is 7.5 times higher than the stick and its shadow is 7.5 times longer.

Answer: The tree is 9 m high.

Example 2: a trip by map

Question: On a map with scale 1 : 50 000 the route to a lookout tower measures 9 cm. How long is it in real life?

Method: The scale says: reality = map · 50 000.

9 · 50 000 = 450 000 cm.

I convert: 450 000 cm = 4 500 m = 4.5 km.

Sense check: 4.5 km is an hour of brisk walking — a reasonable trip.

Answer: The route measures 4.5 km. Scale is a similarity coefficient between paper and landscape; convert units at the end, so you do not get lost in the zeros.

Example 3: the width of a river without getting wet

Question: On the bank you set out a triangle: point A opposite a tree on the other bank, from A you walk 20 m along the bank to point B and from B you aim at the tree. A smaller similar triangle with a helper part has matching sides 4 m (along the bank) and 3 m (at right angles to the river). How wide is the river?

Method: The triangles are similar by AA (a matching angle at B, right angles at the bank).

Coefficient: k = 20/4 = 5.

Width of the river = 3 · 5 = 15 m.

Answer: The river is 15 m wide. The same calculation as with a shadow — only the triangles lie on the ground instead of in the air.

Practice: now you do it 🎲

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

Train converting sides with the coefficient — the core of every shadow problem. Write a whole number.

Practise the questions

On paper (and outside if you like)

Three measuring problems. Sketch, test AA, coefficient, result, sense check.

1. A stick 1 m casts a shadow 2.5 m, a chimney casts a shadow 30 m. Calculate the height of the chimney and check with sense.

2. Map 1 : 100 000: a route measures 7.2 cm. How many km is that in real life? And the other way: how many cm on the map is 15 km?

3. You are 1.6 m and your shadow is 2 m. How long a shadow does a house 12 m high cast at the same moment? (Watch, here you look for the shadow, not the height.)

4. When there is sun: measure the height of something real with a shadow outside and write the whole method as a protocol.

Now you 💪

1. You can explain why shadow triangles are similar (AA: a right angle + the same sun).
2. You can do a map scale both ways: from the map to the landscape and back.
3. You check every result with a sense check.

What to take from this lesson

• Similarity measures what you cannot reach: a shadow carries height, a map distance.
• Shadow method: k = object’s shadow / your shadow, height = your height · k.
• A map scale is a similarity coefficient. Convert units at the end.
• Measure both shadows at the same moment — the sun moves.
• A sense check spots a flipped coefficient at once.

© 2026 Ing. Martin Polak / AlgoRhino · Ketentuan penggunaan konten