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Wikipedia:Reference desk/Archives/Mathematics/2022 December 21

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December 21

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Calculate distance between centers of circles given area of intersections

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I need help checking my work. I'm working on a representation of multiple regression effect sizes using a Venn Diagram for educational policy research. I see that the area of a geometric lens has a closed form solution. Given a Venn diagram made from circles , , and with centers , , and , that , and , then . Can someone validate this? The Venn diagram is represented at this link here.Schyler (exquirito veritatem bonumque) 04:07, 21 December 2022 (UTC)[reply]

Circle has no role. I cannot replicate these numbers. When each circle passes through the centre of the other circle, so the circle centres are apart, a rhombus of two equilateral triangles fits within the lens. The area of this rhombus is so when , contradicting My calculations give me that distance gets you Conversely, to get lens area I find we need  --Lambiam 07:38, 21 December 2022 (UTC)[reply]
Yes, circle has no role. Let be the angle at X (or Y) between the lines to the junctions of ABD and of CEFG. The lens has area , where , and the diagram says (D+G) has area 0.03, so . That gives (in radians). The length from X (or Y) to the midpoint of XY is , and twice this gives . My trig is rusty and I may have made some errors, but I think the principle and the order of magnitude are right. If were 0.23, we'd get and . Certes (talk) 13:31, 21 December 2022 (UTC)[reply]
For we have not quite but at least in the ballpark. But in the second case, you missed a factor : for we have while  --Lambiam 15:52, 21 December 2022 (UTC)[reply]
Here are the calculations for lens area step by step:
--Lambiam 16:03, 21 December 2022 (UTC)[reply]
Oops. Thanks, that looks more credible. Certes (talk) 18:40, 21 December 2022 (UTC)[reply]

Okay, well this is interesting. Yes, in the example, circle C has no role here. Someone said "When each circle passes through the centre of the other circle," but I do not think that is probable. Here were my steps:

Circles , , and have centers , , and and radii . The centers form and intersect such that , ,

Simplifying:

oh I see this was my problem, I think... I distributed the 2 onto

Update: well, here i am checking my own work again. i found another error. i deleted some values of d within . I should know better than to ask for help right away... i can do this... but the doubt is strong in this one

no that's wrong too, ... i got it mixed up in again...

You can shift two equal-sized circles such that each passes through the other's centre; I used this special case merely to easily establish bounds that were violated by a purported solution.
The function is transcendental, and one cannot hope to solve the equation by a combination of algebraic and trigonometric manipulations for rational values of except when is an integer, in which case the equation is solved by  --Lambiam 09:01, 22 December 2022 (UTC)[reply]