A challenging problem of regular heptagon by HC Rajpoot

Geometry Level 4

What is solid angle (in steradian) subtended by a regular heptagonal plane with each side 12 c m 12 cm at a point lying on the perpendicular axis passing through one of the vertices such that its height from the plane is 25 c m 25cm ?


The answer is 0.523322158.

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1 solution

Mark Hennings
Dec 21, 2017

The solid angle subtended by the yellow triangle O A 1 A 0 OA_{-1}A_0 at the point with coordinates ( u , v , 25 ) (u,v,25) is Ω ( u , v ) = 0 6 cot π 7 d x x tan π 7 x tan π 7 25 ( 625 + ( u x ) 2 + ( v y ) 2 ) 3 2 d y \Omega(u,v) \; = \; \int_0^{6\cot\frac{\pi}{7}}\,dx \int_{-x\tan\frac{\pi}{7}}^{x\tan\frac{\pi}{7}} \frac{25}{\big(625 + (u-x)^2 + (v-y)^2\big)^{\frac32}}\,dy Now the solid angle subtended by the complete heptagon at the point with coordinates ( 6 , 0 , 25 ) (-6,0,25) (a point a height of 25 25 above A 3 A_3 ) is equal to the sum of the solid angles subtended by the yellow triangle at the points B j B_j with coordinates ( 6 c o s e c π 7 cos ( 2 n + 1 ) π 7 , 6 c o s e c π 7 sin ( 2 n + 1 ) π 7 , 25 ) (6\mathrm{cosec}\tfrac{\pi}{7}\cos\tfrac{(2n+1)\pi}{7},6\mathrm{cosec}\tfrac{\pi}{7}\sin\tfrac{(2n+1)\pi}{7},25) for 3 j 3 -3 \le j \le 3 (the points which are a height of 25 25 above the points A j A_j for 3 j 3 -3 \le j \le 3 . Thus the desired solid angle is j = 3 3 Ω ( 6 c o s e c π 7 cos ( 2 n + 1 ) π 7 , 6 c o s e c π 7 sin ( 2 n + 1 ) π 7 ) = 0.523322685 \sum_{j=-3}^3 \Omega\big(6\mathrm{cosec}\tfrac{\pi}{7}\cos\tfrac{(2n+1)\pi}{7},6\mathrm{cosec}\tfrac{\pi}{7}\sin\tfrac{(2n+1)\pi}{7}\big) \; = \; \boxed{0.523322685} according to Mathematica using WorkingPrecision set to 20 20 .

A search of the Internet found this formula from the author's book. I used with a = 12 a=12 and h = 25 h=25 , and obtained a different answer... A typo, somewhere, perhaps.

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