Area calculations
Calculate the area of a circle
We all know that for a circle, the formula of area = πr
Now we have a circle as the image shown
Input the amount of pieces:
(bro literally spend 2 hrs to make this)
As the split amount of the circle increasing, the area of reformed shape is getting more and more close to the rectangle.
The width of rectangle is the orginal radius. Because of the cross arrangement sector, the length is half of the circumference.
The formula for circumference is πd, since the circumference is always π times than the diameter (d).
πd divide by 2, then times r is the formula of a circle, simplyfy it:
πd/2*r
=πr*r (since diameter is always 2 times than radius)
=πr
Infinitely close
Now consider the question, is infinitely close to the rectangle equals to the rectangle?
Here is an example:
∵0.333... = 1/3
0.333... * 3 = 0.999..., 1/3 * 3 = 1
∴1 = 0.999...
∴Infinitely close is an equal.
So the area of a circle is exaclty πr
"A farmer uses a circular water sprayer to irrigate a square shaped field. The diameter of the watered circles is 20 metres. "
Calculating the unwatered area
Calculating the unwatered area is easy. All you need to do is to subtract the watered area from the total area.
We know that the watered area is made up of four circles which has the diameter of 20. So the total area as a square is 20 * 20.
Evaluate it. w
=(20 * 2)
=1600-400π
π to the 2nd decimal
=344
Calculating the central shading area
There are multiple ways to evaluate the area of the central shading area.
1st way: only leave the central 1/4 of the pattern, then evaluate.
We can get rid of the outer graph that is necessary as the graph at right shown
As you can see, we have 4 of the 90 degress sector with a radius of 10 meters as the watered area, the whole field has the side length of 20 meters
Now we can evaluate it.
20
=400 - 100π
π to the 2nd decimal
=86
2nd way: calculate all of the unwatered area, then divide.
Look at the graph at the right. You can find the unwatered area can be equally split to 16 pieces of the same shape, and there are total 4 of them in the center.
We we need to do now is to subtract the watered area from the total area to get the unwatered area, then divide by 16 and times 4.
[(20 * 2)
=400-100π
π to the 2nd decimal
=86
Calculating the new central water spiller
Evaluate the center watered circle that is the maximum size but not inheriting others is still easy.
As the image shown on the right, all you need to do is get the diagonal length and subtract those two radius.
The base side length is 20 meters, so the diagonal length is √(20
≈28.28 meters
Reduce the radius of the original circle, you can get the new diameter.
28.28 - (10 + 10) = 8.28 meters
We can get the area with the diameter
πr
=π(8.28/2)
=17.9776π
π to the 2nd decimal
=56.449664
Extension questions
Is the watered area over 80% of the total field?
Original area:
π * 10
π to the 2nd decimal
1312/1600=82%
82%>80% ∴yes
What is the radius of the circle if you want to spill all the field using 4 spiller?
Solve for the diagonal length:
√(20+20)
≈56.57
56.57 / 2 / 2 = 14.1425