What would be the diameter of a circle having a heptagon of sides 45m,60m, 60m,50m,40m,45m and 50m inscribed in it?


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Question Number 56838 by necx1 last updated on 25/Mar/19

$W h a t w o u l d b e t h e d i a m e t e r o f a c i r c l e$ $h a v i n g a h e p t a g o n o f s i d e s 45 m, 60 m,$ $60 m, 50 m, 40 m, 45 m a n d 50 m i n s c r i b e d$ $i n i t ?$
	Answered by MJS last updated on 25/Mar/19

	$I found no exact solution$ $the heptagon consists of 7 triangles$ $the central angle of a triangle with sides r s r$ $is \arccos \frac{2 r^{2} - s^{2}}{2 r^{2}}$ $the sum of tbe central angles is 2 π$ $so we must solve this :$ $\arccos \frac{2 r^{2} - 40^{2}}{2 r^{2}} + 2 (\arccos \frac{2 r^{2} - 45^{2}}{2 r^{2}} + \arccos \frac{2 r^{2} - 50^{2}}{2 r^{2}} + \arccos \frac{2 r^{2} - 60^{2}}{2 r^{2}}) = 2 π$ $\arccos \frac{r^{2} - 800}{r^{2}} + 2 (\arccos \frac{2 r^{2} - 2025}{2 r^{2}} + \arccos \frac{r^{2} - 1250}{r^{2}} + \arccos \frac{r^{2} - 1800}{r^{2}}) = 2 π$ $\Rightarrow r \approx 57.7563 \Rightarrow d \approx 115.513$
	Commented by necx1 last updated on 27/Mar/19

	$h m m m$
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