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Question Number 40686 by MrW3 last updated on 26/Jul/18

Commented by MrW3 last updated on 26/Jul/18

$F i n d r a d i u s R o f t h e c i r c u m s p h e r e o f$ $p y r a m i d .$ $I g o t R = \frac{\sqrt{3} b^{2}}{2 \sqrt{3 b^{2} - a^{2}}}$
Commented by ajfour last updated on 26/Jul/18

Commented by ajfour last updated on 26/Jul/18

$R^{2} = x^{2} + \frac{a^{2}}{3} . . . (i)$ $b^{2} = \frac{a^{2}}{3} + {(R + x)}^{2} . . . . (i i)$ $f r o m (i i) x = [\frac{\sqrt{3 b^{2} - a^{2}}}{\sqrt{3}} - R]$ $u s i n g t h i s i n (i)$ $R^{2} = {[\frac{\sqrt{3 b^{2} - a^{2}}}{\sqrt{3}} - R]}^{2} + \frac{a^{2}}{3}$ $\Rightarrow \frac{2 R}{\sqrt{3}} (\sqrt{3 b^{2} - a^{2}}) = b^{2}$ $\Rightarrow R = \frac{b^{2} \sqrt{3}}{2 \sqrt{3 b^{2} - a^{2}}} .$
Commented by MrW3 last updated on 26/Jul/18

$t h a n k s s i r!$ $I t r i e d i n t h i s w a y :$ $h = \sqrt{b^{2} - {(\frac{a}{\sqrt{3}})}^{2}}$ $\frac{2 R}{b} = \frac{b}{h}$ $\Rightarrow R = \frac{b^{2}}{2 h} = \frac{b^{2}}{2 \sqrt{b^{2} - \frac{a^{2}}{3}}} = \frac{\sqrt{3} b^{2}}{2 \sqrt{3 b^{2} - a^{2}}}$
Commented by ajfour last updated on 26/Jul/18

$I t s b e t t e r S i r!$
Commented by MrW3 last updated on 26/Jul/18

$T h a n k y o u s i r!$ $I s t h e r e a w a y t o c a l c u l a t e R f o r a n y$ $p y r a m i d w i t h s i d e s a, b, c (b a s e) a n d$ $p, q, r ?$
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