Step by step
The question
Question: find the area of a circle with radius 6.5 cm.
1 Use the rule
Walk once around any circle.
It is a bit more than 3 times the way across.
It is about 3.14 times.
This number is called π.
So the way around is π times the way across: C = π × d.
We use this to see why the area rule works.
Why is the area π times r squared?
Cut the circle into thin slices, like a pizza.
Lay the slices in a row.
Turn every other one upside down.
It is almost a rectangle.
Its height is one slice: the radius, r.
Half of the slices have their round edge on top.
So the top is half the way around.
The way around is π × d.
d is 2 times r.
So half the way around is π × r.
Rectangle: length times height.
So the area is π × r times r.
Thinner slices make the edges flatter, so it gets closer to a real rectangle.
r times r is r squared.
So A = π × r².
Now put the numbers in: r = 6.5.
6.5² means 6.5 times 6.5.
Forget the points: 65 times 65 is 4225.
6.5 is 65 tenths.
6.5 is 65 tenths.
Tenths times tenths make hundredths.
That is 2 digits after the point: 42.25.
So A = π × 42.25.
We write it as 42.25π.
We keep π in the answer.
That is the exact answer.
2 About how much
To see about how much, put 3.14 for π.
Forget the points: 4225 times 314 is 1326650.
42.25 is 4225 hundredths.
3.14 is 314 hundredths.
Hundredths times hundredths make ten-thousandths.
That is 4 digits after the point: 132.6650, which is 132.665.
That is 132.665.
Rounded to 2 decimal places, it is 132.67.
So the area is about 132.67 cm².
Let's look back.
First: use the rule.
Last: about how much.