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Geometry/Segment of a Circle


QUESTION: sir. i actually have an answer instead of a question. hope you can guide me. i have been able to find the area of the segment of a circle without using trigonometry, where only the radius of circle is given along with the angle subtended by the chord to the centre. till now ive been able to find it for quite a number of angles and soon i hope to find a uniform method to find the area for almost any angle.the thing is that one of my mathematically inclined friend had asked me to do this work.he said that using this he could somehow derive a proper formula for "sin theta" which would somehow be big thing in math.he even claimed to share the "fields medal" with me! now he says that he just is not able to produce the formula that he had promised and since the start he wont tell me how he would derive the formula. this incident made me think weather my work is really of some importance.i planned to send it to an organization which gives awards to math innovations but i fear weather it is just simple math. i mean the other projects in that org are so cool with words like "dimensions and matrixes" and mine is just "geometry of circle". sir please i would like you to tell me if my work is of any utility. that org is ""

ANSWER: Hi Siddhant,

The area of a segment of a circle can be found using only the radius and the angle to the centre.  Find the area of the full circle, then multiply that by the angle over the full circle.  So if you had radius "r" and angle of "d" degrees, the segment's area would be πrē(d/360).

Just want to make sure I am not misunderstanding the scenario you describe.  Feel free to ask a follow-up with a diagram.  I would be happy to discuss this with you further.


---------- FOLLOW-UP ----------

sector and segment
sector and segment  
QUESTION: sir i think what u are talking about is the sector of a circle and not segment of a circle. sector of a circle is the area of circle multiplied with the angle over 360.i am talking about the "segment of a circle" for which the existing formula includes trigonometry and i have derived method to find area of it without trigonometry. so is it of some work?

Hi Siddhant,

Thanks for clarifying.  You asked if it is just "simple math" and my answer to that is yes.  The result (determining the area of a segment) can be readily achieved using trigonometry.  No particularly involved mathematics are required in solving this problem.  As it stands, calculating the area of a segment is pretty simple, so it is unlikely that you have simplified it much further.  It is also possible that the result you have found was previously found (e.g. is older than trigonometry).

That said, the result you claimed to have derived is certainly an interesting one!  I am definitely curious as to how you have done it.  Perhaps there is something fascinating and elegant you have derived, but I would have to see it to comment further.

Thanks for asking,


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Azeem Hussain


I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.


4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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