A ring of ice and stone

Through the window shone the ring of Saturn – a flawlessly round band of ice and chunks of rock that surrounds the planet. Liri stared at it for a long time. “Every point of this ring seems to be exactly the same distance from Saturn’s centre… just like a round window that I once knew.”

Erklärus: “That is the definition of a circle – and you have just discovered it yourself. Let’s reconstruct what else you know.”

⭕ Centre, radius and diameter

A circle is the set of all points that are exactly the same distance from a fixed centre M. This distance is called the radius r.

Liri: “Like a pair of compasses! Point in the middle, and turn it round once.”
Erklärus: “Exactly that picture. And a straight line through the centre that touches the circle on both sides is the diameter d – always twice as long as the radius: d = 2 × r.”
M r d = 2r
Labels in the circle
Centre M
The same distance from every point on the circle
Radius r
Distance from the centre to the edge
Diameter d
d = 2 × r
Radius = 6 cm. How big is the diameter?

🔢 The secret of π

Liri: “You mentioned π. What is that?”
Erklärus: “For every circle in the world – no matter how big – measure the circumference and divide it by the diameter. You always get exactly the same number: π ≈ 3.14159…”
Liri: “The same number for every circle? I can hardly believe that!”
Erklärus: “Because the circumference and the diameter always grow in the same ratio. π is irrational – the decimal places never stop, with no pattern.”
The number π
π ≈ 3.14159265…

π = circumference ÷ diameter. For school sums π ≈ 3.14 is enough.

📏 The circumference

Erklärus: “From π = C ÷ d it follows directly: C = π × d. And because d = 2r, it is also true that C = 2 × π × r.”
Formula – circumference
C = π × d = 2 × π × r
Radius 5 cm – what is the circumference? (π≈3.14, to 1 decimal place)

🟡 The area – why r² is hidden in the circle

Liri: “The circle has no straight sides – so how do you work out the area?”
Erklärus: “A trick: cut the circle into lots of pieces like an orange and put them back together, alternately with the point up and down. The more pieces, the more the result looks like a rectangle.”
→ π × r (= C/2) r

The “slices of cake” make a rectangle: width r, length π×r.

Formula – area
A = π × r²
Radius 5 cm – what is the area? (to 1 decimal place)

🧮 Liri’s circle calculator – interactive

Enter a radius – the circumference and area appear straight away:

⭕ Calculate a circle
Radius r = cm
🌌 Circles in the universe

Planets and stars are almost perfect spheres – their cross-section is a circle, shaped by their own gravity. Most orbits are also (nearly) circular.

❌ Common mistake

The radius and the diameter are often swapped! Always check: the radius goes from the centre to the edge, the diameter from edge to edge (through the centre).

📝 Exercise 1
Diameter 20 cm – how big is the radius?
📝 Exercise 2
Radius 3 cm – what is the circumference? (to 1 decimal place)
📝 Exercise 3
Radius 4 cm – what is the area? (to 1 decimal place)

A round window found again

Liri now drew the round window onto her star map to scale – with exact radius, circumference and area.

Liri: “Now I know exactly how big my window was – and I can recognise it on the map when I see it.”
Erklärus: “Every memory that is worked out makes your map more reliable.”
“Circle: r=radius, d=2r. C=π×d=2πr. A=π×r². π≈3.14159, irrational.”
What you learned today
A = π × r²
The circumference grows linearly with r, the area grows quadratically
🌟

Chapter 50 mastered!

Liri can now work out the circumference and area of every circle.

Erklärus’s notebook – page 50:
“π connects the circumference and the diameter in every circle. C=2πr, A=πr².”
← Back to overview 50 – The Circle