Prairie Minds

In this lesson

What is circle area?

Fly over southern Alberta in July and you'll see them: enormous green circles stamped onto the prairie, each one drawn by a pivot arm sweeping a slow lap around a well. The farmer doesn't care much how far the wheels travel around the edge — last lesson's formula covers that. The question that pays the bills is how much land is inside the circle. And the edge formula is no help at all.

Circumference is the distance around the circle:

C = \pi D

Area is the amount of surface inside it — measured in square units, such as cm^2, m^2, or km^2, because it's a different kind of size. The inside needs its own formula, and you're about to build it out of pieces of the circle itself.

Rearrange the circle

Cut a circle into thin sectors and stack them tip-up, tip-down, tip-up. The lumpy result starts hinting at a shape you already know how to measure. Slice thinner until you're sure what it's becoming — then call its base and height before the labels appear.

Try it

Slice the circle thinner and watch what the pieces settle into.

Number of sectors

n = 6

Still lumpy. Thin the slices and look again.

The pieces never stopped being the circle — same surface, new outline. So the parallelogram's area IS the circle's area:

A = \text{base} \times \text{height}

A = \pi r \times r

A = \pi r^2

That's the circle area formula, and the square is no mystery: it comes from multiplying one length by another length. The full circumference is $2\pi r$ , and only half of it lies along the base — the other half runs along the top — which is where the $\pi r$ comes from.

Calculate area

Use $\pi \approx 3.14$ .

If a circle has radius 3 cm:

A = 3.14(3^2)

A = 3.14(9)

A = 28.26\text{ cm}^2

If the question gives diameter first, halve it to get the radius before using the area formula.

For diameter 20 cm:

r = 20 \div 2 = 10

A = 3.14(10^2) = 314\text{ cm}^2

Where it shows up in real life

Back to the crop circles from the opening. The pivot arm is the radius made physical — a 400 m pipe anchored at the well, sweeping the full lap.

If the irrigated radius is 400 m, the watered area is:

A = 3.14(400^2)

A = 502{,}400\text{ m}^2

That is about 50.24 hectares, since one hectare is 10,000 m^2.

Misconception probe

Worksheet

These are not graded. Keep asking: am I measuring the outside edge or the inside surface?

Practice · Not graded

MA.7.MEA.1

01 / 08

Which formula finds the area of a circle?

Going further

The next circle lesson mixes problems in both directions. Sometimes you will find circumference, sometimes area, and sometimes you will work backward from one measurement to find the radius or diameter first.