Three squares, one triangle

Erklärus showed Liri how navigators sight two stars at right angles to each other to work out their own position. “I know this method!” Liri suddenly cried. “At home the technicians always drew three squares on the sides of a strut before they built anything – to check whether an angle was really exactly 90°.”

Erklärus: “That is Pythagoras’ theorem – one of the most famous formulas in mathematics. Let’s reconstruct how it works.”

📐 Only for right-angled triangles

In a right-angled triangle the two shorter sides are called the legs (a, b) and the longest side, opposite the right angle, is the hypotenuse (c).

b a c
Liri: “How are a, b and c connected?”
Erklärus: “That is exactly what Pythagoras’ theorem says: the square on the hypotenuse is exactly as big as the two squares on the legs together.”

🔲 The proof with three squares

Draw a square on each side of the right-angled triangle. The area of the big square (on c) is exactly as big as the two small squares (on a and b) together.

a² = 9 b² = 16 c² = 25 a = 3 b = 4 c = 5

a² + b² = c² → 9 + 16 = 25 – the areas of the small squares add up to the area of the big one.

Pythagoras’ theorem
a² + b² = c²

Only valid for right-angled triangles. c = hypotenuse (longest side, opposite the right angle).

🧮 Using it – working out the missing side

Erklärus: “a=3, b=4 – how long is c?”
Liri: “3² + 4² = 9+16 = 25. And c = the square root of 25 = 5!”
Erklärus: “Perfect – the famous 3-4-5 triangle. If you want to work out a leg, you simply rearrange: a² = c² − b².”
Rearranged

c = √(a²+b²)  ·  a = √(c²−b²)  ·  b = √(c²−a²)

a=6, b=8 – how long is c?

🧮 Liri’s Pythagoras calculator – interactive

📐 The hypotenuse from two legs
🌌 Pythagoras in the universe

Navigation systems (including those of space probes) use Pythagoras’ theorem all the time, to work out the direct distance from two perpendicular distances (e.g. east–west and north–south).

❌ Common mistake

The theorem applies only to right-angled triangles! And the hypotenuse c is always the longest side, opposite the right angle – not just any side.

📝 Exercise 1
a=5, b=12 – how long is c?
📝 Exercise 2
c=10, a=6 – how long is b? (b=√(c²−a²))
📝 Exercise 3
a=9, b=12 – how long is c?

The right angle confirmed

Liri could now check for herself whether a strut was really at a right angle – and wrote the method onto her star map.

Liri: “With three simple numbers I can check whether an angle is exactly 90° – without measuring!”
Erklärus: “That is exactly what makes Pythagoras so powerful. And now you remember yet another tool from your home.”
“Pythagoras’ theorem: a²+b²=c², only for right-angled triangles. c=hypotenuse (longest side, opposite the right angle).”

The place on the map

At the end Erklärus spread out Liri’s star map once more. For 54 chapters they had entered it point by point: every memory a coordinate, every distance worked out. Three points stood out – the beacon, the shipyard and the round window.

“Let’s use Pythagoras to check whether this triangle has a right angle,” he said. He calculated – and the squares on the two short sides added up to exactly the square on the long one. “A right angle! And in the whole sky there is only one place where three bright stars form a triangle like that.”

He pointed the telescope to the west. “Vega, Deneb and Altair – the Summer Triangle. It is roughly right-angled. And at the right angle sits the brightest of them: Vega, in the constellation Lyra, the Lyre.”

Vega (Lyra) Deneb (Cygnus) Altair (Aquila) ≈ 90°

The Summer Triangle (schematic, not to scale)

Liri had gone quite quiet. She shone in a cool blue-white.

Erklärus: “You knew. All along.”
Liri: “Yes. I am Vega – the brightest star of the Lyre. I always knew who I was. It was only the way back that I didn’t know. And I didn’t want to give away this big secret – I wanted you to find it out yourself. At home everyone calls me Liri. I have got so used to it that it feels more real than Vega.”
Erklärus: “And all the memories you told me about…”
Liri: “…were clues. You read every single one of them correctly.”
Erklärus: “Then in the last chapter let’s look back once more at everything we have been through together – before you go home.”
“The Summer Triangle is roughly right-angled. At the right angle shines Vega, the brightest star in the constellation Lyra – Liri’s home. She knew it from the very beginning.”
What you learned today
a² + b² = c²
Only for right-angled triangles – c is always the hypotenuse
🌟

Chapter 54 mastered!

Liri can now work out every missing side of a right-angled triangle.

On to the grand finale: Chapter 55 →
Erklärus’s notebook – page 54:
“a²+b²=c² only applies to right-angled triangles. c is always the longest side.”
← Back to overview 54 – Pythagoras’ Theorem