Shall we find the area of the circle?
You are going round and
round wondering how to find the area of a circle. It's a circle so it's no wonder
you're thinking and going in round and round. So I hope you are very excited
about seeing the area of the circle today.
Before that recall the
formula for the circumference of a circle. You ask me whether you will forget
what you saw yesterday. I know you won't forget. I told you to remember it
because the formula for the circumference of a circle is going to help us find
the area of a circle.
I know by now you have
remembered that the circumference of a circle = 2 πr units.
Also recall the formula
for the area of parallelogram. You ask, don't we know it's bh square units? Can math
formulae you don't know exist in math anymore?
Now let's find the
formula for the area of a circle?
Does a circle have no
sides? If there are sides, can you multiply the adjacent sides to give an easy
formula for the area? What do we do now?
Draw a circle and draw
as many diameters as you can into it.
When drawing the
diameters, draw them so that they divide the circle equally. For example, when
drawing a diameter, the circle should be divided into two equal parts. When two
diameters are drawn, they should be divided into four equal parts. When drawing
four diameters, divide them into eight equal parts. If eight diameters are
drawn, they should be divided into sixteen equal parts. Sixteen diameters
should be drawn into thirty-two equal parts. Draw as many diameters as you can
within a circle so that they are divided into equal parts.
As you increase the
diameters, the figure for the area becomes precisely a parallelogram. How can
it be parallelogram you ask?
Now you cut the circles
through the diameters into pieces and glue them as shown in the picture.
At most it is enough if
you draw 16 diameters and divide them into 32 equal parts. Now you have 32
equal parts of the circle. The length of these equal parts is equal to the
radius and the width at the top is significantly less than the circumference of
the circle. Paste these parts as shown in the picture.
After you glue the
pieces of the circle looks like a parallelogram. Do you know the formula for
the area of a parallelogram? We remember it as bh square units.
Now, isn't the base b
of the parallelogram that we stick together with the pieces of the circle half
the circumference of the circle! If it is half, then where is the other half,
you ask? It is on the upper side of the parallelogram. That is, one half of the
circumference of the circle is below and the other half of the circumference is
above. So the base of the parallelogram is half the circumference of the
circle. That is 2 πr / 2 = πr units
Next the distance
between the bottom and the top of the parallelogram i.e. the height of the
parallelogram h is the radius as it is the distance from the center of the
circle to the circumference of the circle. That is, h = r.
Tell me now.
Area of parallelogram = bh
square units Substituting πr for b and r for h gives πr × r = πr2 square units right! This is the formula
for the area of a circle. What do you understand?
Tomorrow we will also
look some more things at the area of a circle, because there are many things to
know about the area of a circle.
*****
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