Circumference of a circle (π) · Lập trình với Martin

Meeting the number π and the formula o = π · d. How far does a wheel go in one turn?

The number hiding in every wheel

Take a tape measure and measure the circumference of a mug. Then its diameter. Divide. You will get something around 3.1. Do it with a plate, a bucket, a tyre — every time roughly 3.14.

This number is famous π (read “pi”). The circumference of every circle is π times longer than its diameter. Every one. A small coin and a giant Ferris wheel.

π = 3.14159… — the decimal never ends and never repeats. We will calculate with π ≈ 3.14, which is plenty for school problems.

The formula for circumference: o = π · d, or through the radius o = 2 · π · r (because d = 2r).

What is it for? How much fence around a round flower bed. How far a wheel goes in one turn. How long a string around a tree trunk is. Circles are everywhere — and today you will measure their circumference for the first time without a tape.

Jonáš has a new bike and on it a computer measuring speed. When setting it up it asks for “tyre circumference in millimetres”. Jonáš is stuck — a tape around the tyre will not hold. Filip: “Measure the diameter, that is a straight line.” Diameter including the tyre: 66 cm. “And now times pi: 66 · 3.14 = 207.24 cm. Rounded 2072 millimetres.” They enter it into the computer and the speed matches the phone in his pocket. “So in one push on the pedals I go two metres?” — “In one turn of the wheel. Exactly.”

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In the question watch whether you have a radius or a diameter. The formula o = π · d wants the diameter; if you get a radius, double it first. Half the mistakes with circles are swapping r and d.

How to calculate the circumference

Formula: o = π · d = 2 · π · r. Both versions are the same — pick according to what you know.

Method:

Find out what you have: radius or diameter?
Substitute into the formula with π ≈ 3.14.
Multiply and round reasonably (to tenths or whole numbers).
Write the unit — circumference is a length: cm, m, km.

Example: a plate with diameter 24 cm. o = 3.14 · 24 = 75.36 cm, rounded 75.4 cm.

The other way — diameter from circumference: a tree has trunk circumference 157 cm. What is the diameter? I turn the formula (lesson 19!): d = o/π = 157 : 3.14 = 50 cm. That is how foresters measure the thickness of trees: circumference with a tape, diameter by calculating.

Estimate without a calculator: π is roughly 3, so the circumference is roughly triple the diameter. Diameter 24 cm → circumference a bit over 72 cm. If 7 cm or 700 cm comes out, the point ran away — the estimate catches it.

Example 1: a fence around a flower bed

A round flower bed has radius 1.5 m. How many metres of fence do you need?

I have a radius, I use o = 2 · π · r:

o = 2 · 3.14 · 1.5 = 3.14 · 3 = 9.42 m.

I will buy 10 m of fence — a reserve for joins.

Estimate as a check: the diameter is 3 m, circumference roughly triple, so a bit over 9 m. It fits.

Example 2: how many turns a wheel makes

A wheel has diameter 66 cm. How many times does it turn on a path 1 km long?

Circumference = one turn: o = 3.14 · 66 = 207.24 cm ≈ 2.07 m.

Path: 1 km = 1000 m.

Turns: 1000 : 2.07 ≈ 483 turns.

Two-step problems are typical with circles: first the circumference, then something else with it. Split it and write mid-results.

Example 3: diameter from circumference

A runner went around a round pond and the watch showed 94.2 m. What is the diameter of the pond?

I turn the formula o = π · d into d = o/π:

d = 94.2 : 3.14 = 30 m.

And the radius r = 15 m.

Check with the forward formula: 3.14 · 30 = 94.2. It fits. Making a quantity the subject of a formula (lesson 19) meets geometry here — everything you have learned fits together.

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Do not round π to 3 — the difference is almost 5% and with a fence for thousands of crowns it shows. Calculate with 3.14 and round only the RESULT. And do not forget: o = π · d wants the diameter, not the radius.

Practice: now you do it 🎲

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

Calculate the circumference of a circle. Calculate with π ≈ 3.14 and write decimals with a point.

Practise the questions

On paper

Measure, calculate, check with string. Today you will feel π at home.

Measure the diameter of a mug with a ruler. Calculate the circumference with π ≈ 3.14. Then wrap string around the mug, measure the string and compare with the calculation.
Calculate the circumference of a circle with radius 5 cm and the circumference of a circle with diameter 5 cm. Write why they came out different.
A wheel has diameter 70 cm. Calculate the circumference and how many turns it makes in 500 metres.
The circumference of a round table is 3.77 m. Calculate the diameter and decide whether the table fits into a corner 1.5 m wide.

Now you 💪

I know the formula o = π · d = 2πr and I know when to use which version.
I calculate with π ≈ 3.14 and I round only the result.
I can calculate the diameter back from the circumference: d = o/π.

✅

Done when: You have a mug measured and checked with string, circumferences calculated and a diameter back, and a run of five correct in practice.

What to take from this lesson

The circumference of a circle is π times the diameter: o = π · d = 2πr.
π ≈ 3.14 — an endless number that is the same for every circle.
Watch the difference between radius and diameter.
Back from circumference: d = o/π.
A quick estimate: circumference is roughly triple the diameter.