Step by step
The question
Question: find the area of a circle with radius 7.
There are four choices.
First we solve it, the normal way.
Then we pick.
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.
7² means 7 times 7.
That is 49.
So A = π × 49.
We write it as 49π.
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: 49 times 314 is 15386.
3.14 is 314 hundredths.
So the answer is 15386 hundredths.
That is 2 digits after the point: 153.86.
So the area is about 153.86 square units.
3 Pick the choice
Our answer is A ≈ 153.86.
That is choice C.
4 Why not the others
A is 21.98.
Wrong: π × 7: used r, not r².
B is 43.96.
Wrong: 2πr: that is the circumference.
D is 615.44.
Wrong: used the diameter 14 in πr².
Let's look back.
First: use the rule.
Then: about how much.
Then: pick the choice.
Last: why not the others.