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
Calculate
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.

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Find the length of one arc that a sector by multiplying a radius by an edge of a ar in radians.

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Find the length of a chord of a ar by using law of cosines (a chord is the basic of the isosceles triangle, with two radiuses together legs and also sector angle as apex angle).

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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.


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Cutting a Circle right into Three parts with two Parallel Lines

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.


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y=f(x)

Using the Pythagorean theorem, the y duty is

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To discover an area of a left slice, you need to integrate this role from -R come x. The antiderivative the our duty is :

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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

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, hence

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Our final integral is

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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)

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Thus we deserve to equate

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Which gives us

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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

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, three times much more for third point
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and also so on.

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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