Area of a Circle

The area of a circle is π × r × r, written πr².

It is the radius that is squared

Given a diameter of 10, halve it first. Squaring 10 instead of 5 gives four times too much.

Square before multiplying

πr² means square the radius, then multiply by π. Not (πr)².

Doubling the radius

Double the radius and the area becomes four times bigger, because the radius is used twice.

Why the radius is squared

Cut a circle into many thin wedges and rearrange them alternately point-up and point-down. They form something close to a rectangle of height r and width half the circumference, πr. Its area is πr² — the formula, derived.

What the squaring means practically

Double the radius and the area quadruples. A 12-inch pizza has four times the area of a 6-inch one, not twice. This is the scale-factor squaring from Chapter 1, and it is why the larger pizza is usually much better value.

Estimating a circle's area

Since π is a bit over 3, the area is a bit more than three times the radius squared. For r = 5, that is a bit over 75 — the exact value is about 78.5. This estimate is quick and catches a misused diameter at once.

Half and quarter circles

A semicircle has half the area, a quarter circle a quarter. Find the whole circle's area first and then take the fraction — trying to adjust the formula itself is where errors creep in.