Step by step
The question
Question: find the area of a circle with radius 7.5 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 = 7.5.
7.5² means 7.5 times 7.5.
Forget the points: 75 times 75 is 5625.
7.5 is 75 tenths.
7.5 is 75 tenths.
Tenths times tenths make hundredths.
That is 2 digits after the point: 56.25.
So A = π × 56.25.
We write it as 56.25π.
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: 5625 times 314 is 1766250.
56.25 is 5625 hundredths.
3.14 is 314 hundredths.
Hundredths times hundredths make ten-thousandths.
That is 4 digits after the point: 176.6250, which is 176.625.
That is 176.625.
Rounded to 2 decimal places, it is 176.63.
So the area is about 176.63 cm².
Let's look back.
First: use the rule.
Last: about how much.