Question indigenous Henry:Where execute you ar a chord in a one such that it divides the areaof the circle 1/3 and also 2/3? Also, what is the length of the chord?

Hi Henry.

You are watching: 1/3 of a circle

You are describing a circular segment who area is (1 /3 )r2 , yet I doubt you currently know that.

The area of a one segment have the right to be deduced native the area the the one sector and also the area of the isosceles triangle within the circular sector. This chart helps describe things:

 minus equals A=½r2 A=½r2sin A=½r2(-sin)

Thus, you desire to deal with (1 /3 )r2 = ½r2(-sin), i m sorry simplifies come 3 - 3sin - 2 = 0. Note that the angle is the central angle measure in radians.

This equation is not basic to solve. In fact, the easiest approach is to fix it numerically (that is, to discover a close approximation).

Using a spreadsheet package, i calculated an approximation the f() = 3 - 3sin - 2.

Clearly, have to be in between zero and also radians, so using those together the initial boundaries, ns quickly split the selection of feasible values the in fifty percent and zeroed in top top the approximate worth of that answers her question.

So

2.6053 radians. Thus, a circular segment subtended by an angle of 2.6053 radians has an area the one 3rd of the area that the one itself.

The length of the chord can be calculated given the radius r and also this angle

using an easy triangle geometry:

 implies

so sin( / 2) = (c / 2) / r.

Hope this helps, Stephen La Rocque.

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Math main is sustained by the university of Regina and also The Pacific Institute for the math Sciences.