Below friend can discover two calculators i m sorry calculate exactly how to reduced a circle into equal components - traditional and also non-traditional way. By the timeless way, ns assume cutting a circle right into sectors, similar to you usually reduced a pie or pizza. And by the non-traditional way, i assume cutting a circle right into equal upright slices through parallel currently or with parallel chords, if girlfriend like. Both calculators existing a drawing that illustrates the result. And also you can uncover all formulas and math in the post below the calculators.
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Cutting a Circle right into equal sectors
Circle Radius
Number of Sectors
Calculation precision
Digits ~ the decimal point: 2
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Angle that a Sector
Length of one Arc the a Sector
Length of a Chord of a Sector
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Cutting a Circle into equal components with Parallel Cuts
Circle Radius
Number of Parts
Calculation precision
Digits ~ the decimal point: 2
Calculate
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Cutting a Circle right into Sectors
Ok, you require to cut a circle right into several sectors (even non-even numbers). To perform this, you require to find the parameters that a sector. The is a simple task:
Find the edge of a sector in radians by dividing 2π (representing 360 levels in radians) through a variety of sectors.
This totally defines all N same sectors.
Cutting a Circle with Parallel Cuts
This means is more interesting. For simplicity, I will consider half of a circle because it is symmetrical.
Let"s part it with vertical slices. In this case, we require to uncover the x-coordinates that parallel chords, which should separation our circle into equal-area parts. (see clues x1 and x2 ~ above the photo above). Let"s derive the basic formula because that an area that a left slice.
Our half-circle can be assumed of as a role y=f(x), where x - is the coordinate follow me the abscissa axis, and also y is the role equal to the value of the corresponding half-circle point.
Using the Pythagorean theorem, the y duty is
To discover an area of a left slice, you need to integrate this role from -R come x. The antiderivative the our duty is :
We require to uncover the value of constant. Obviously, in ~ the allude where x amounts to -R area must be zero. If us plug -R instead of x right into the formula above, we gain
Our final integral is
Now exactly how do we discover x that the an initial cut? We understand we area us should get - Nth component of the total area (note the half-circle)
Thus we deserve to equate
Which gives us
This is a transcendental equation, and we need to use numerical approaches to deal with it, because that example, Bisection an approach or Newton"s method. Below I used Newton"s method.
The next points of cut can be discovered with the same approach. We require to cut two times an ext for 2nd point
See more: How Many Bumps On A Basketball (Official Rules), How Many Bumps On A Basketball
Then us can find all other parameters, choose chord length, utilizing the allude coordinates.
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