Area of a circle · 跟马丁学编程

The formula S = π · r²: how much area fits into a circle and why pizza size grows faster than the diameter.

How much fits into a circle

You have circumference. Now area — how much space a circle takes. How much grass for a round flower bed, how much dough for a pizza, how much fabric for a round tablecloth.

The formula: S = π · r². Pi times the radius squared.

Watch the order: square the radius first, then multiply by π. r² is a power from lesson 1 — and it must be squared before you multiply. Anyone who calculates (π · r)² gets a completely different (wrong) number.

In the formula there is r, not d. If the question gives a diameter, halve it first.

And one thing that surprises: because there is a square in the formula, area grows much faster than the radius. Twice the radius = four times the area. That is why a large pizza almost always pays more than two small ones — and today you will calculate it.

Ema and Filip are ordering pizza. A small one (diameter 26 cm) costs 169 Kč, a large one (diameter 40 cm) 299 Kč. Filip: “We’ll take two small ones, that is more pizza for almost the same money.” Ema pulls out a phone with a calculator: “Small: radius 13, area 3.14 · 169 = 531 cm². Two small: 1062 cm². Large: radius 20, area 3.14 · 400 = 1256 cm².” Filip stares: the large pizza has more dough than TWO small ones — and costs less than they do. “The square,” Ema shrugs. “Area grows faster than it looks.”

💡

Order of calculation: first r², then times π. With radius 5: first 25, then 3.14 · 25 = 78.5. Never multiply 3.14 · 5 and then square — that is a different and wrong calculation.

How to calculate the area

Formula: S = π · r².

Method:

Find the radius. Diameter in the question? Halve: r = d/2.
Square: r².
Multiply by π ≈ 3.14.
Unit: area is in cm², m² — always “squared”.

Example: a circle with radius 4 cm. r² = 16, S = 3.14 · 16 = 50.24 cm².

The other way — radius from area: a round rug has 12.56 m². What is the radius? I turn the formula: r² = S/π = 12.56 : 3.14 = 4, so r = √4 = 2 m. Square root from lesson 2 in action.

When circumference and when area? Ask: am I measuring a LINE around (tape, fence, string), or AREA inside (paint, grass, dough)? Line = circumference (units without a square), area = area (units squared). The unit of the result always gives you away.

Estimate: π is about 3, so area is roughly three squares r × r. Radius 10 → roughly 300, exactly 314. The estimate catches lost decimal points.

Example 1: grass for a round flower bed

The flower bed has diameter 6 m. How many m² of lawn do you buy?

Diameter → radius: r = 6 : 2 = 3 m.

I square: r² = 9.

I multiply by π: S = 3.14 · 9 = 28.26 m².

I will buy a pack for 30 m².

Estimate as a check: three squares 3 × 3 = 27. 关闭 to 28.26 — it fits.

Example 2: a large pizza, or two small ones?

Small pizza d = 26 cm, large d = 40 cm. How many times more dough does the large have?

Small: r = 13, S = 3.14 · 169 = 530.66 cm².

Large: r = 20, S = 3.14 · 400 = 1256 cm².

Ratio: 1256 : 530.66 ≈ 2.4 times more.

The diameter grew only 1.5 times (from 26 to 40), but the area 2.4 times — because the radius is squared in the formula. A small difference in diameter, a big one in food.

Example 3: radius from area

A round mirror has area 0.5024 m². What is its radius and diameter?

I turn the formula: r² = S/π = 0.5024 : 3.14 = 0.16.

I take the square root: r = √0.16 = 0.4 m.

Diameter: d = 0.8 m.

Check forwards: 3.14 · 0.4² = 3.14 · 0.16 = 0.5024. It fits. The whole of year 8 in one example: a formula, making a quantity the subject, a square root.

⚠️

Into the formula S = π · r² belongs the RADIUS. If you substitute the diameter, the area comes out four times bigger — with a flower bed you would buy four times the grass. A diameter in the question always first halved.

Practice: now you do it 🎲

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

Calculate the area of a circle. First r², then times π ≈ 3.14. Write decimals with a point.

Practise the questions

On paper

With every problem first decide: a line (circumference), or an area (area)?

Calculate the area of a circle with radius 6 cm and the area of a circle with diameter 6 cm. Explain the difference.
A round tablecloth should have diameter 1.8 m. How many m² of fabric is needed? And how many metres of trim for the edge? (Two questions — one for area, the other for circumference.)
Recalculate the pizza problem from the scene with prices from your favourite place (find them in a leaflet or on the web with a parent). Does the large one pay?
A round fountain has area 28.26 m². Calculate the radius (turn the formula and take the square root).

Now you 💪

I calculate in order: first r², then times π.
A diameter from the question I always halve first.
I tell apart when a problem wants circumference (a line) and when area (a surface).

✅

Done when: You have areas calculated and a combined tablecloth problem (area + circumference), your own pizza numbers, and a run of five correct in both practices.

What to take from this lesson

Area of a circle: S = π · r². Power first, then multiplying.
In the formula is the radius — first halve the diameter.
Twice the radius = four times the area.
A line around = circumference (m), area inside = area (m²).
Back from area: r = √(S/π).