Question-40581

Question Number 40581 by ajfour last updated on 24/Jul/18

Commented by ajfour last updated on 24/Jul/18

F i n d r a d i u s o f i n s c r i b e d s p h e r e

w h i c h i s t a n g e n t t o a l l f a c e s o f

t h e t r i a n g u l a r p y r a m i d .

Commented by ajfour last updated on 24/Jul/18

S e e Q . 40594 f o r a n a l t e r n a t e s o l u t i o n .

Answered by MrW3 last updated on 24/Jul/18

h e i g h t o f p y r a m i d :

h = \sqrt{b^{2} - {(\frac{2}{3} \times \frac{\sqrt{3} a}{2})}^{2}} = \sqrt{b^{2} - \frac{a^{2}}{3}}

V = \frac{1}{3} \times \frac{a}{2} \times \frac{\sqrt{3} a}{2} \times \sqrt{b^{2} - \frac{a^{2}}{3}} = \frac{a^{2} \sqrt{3 b^{2} - a^{2}}}{12}

S = \frac{a}{2} \times \frac{\sqrt{3} a}{2} + 3 \times \frac{a}{2} \times \sqrt{b^{2} - \frac{a^{2}}{4}} = \frac{a (\sqrt{3} a + 3 \sqrt{4 b^{2} - a^{2}})}{4}

\Rightarrow r = \frac{3 V}{S} = \frac{a \sqrt{3 b^{2} - a^{2}}}{\sqrt{3} a + 3 \sqrt{4 b^{2} - a^{2}}}

Commented by MrW3 last updated on 24/Jul/18

L e t M b e t h e c e n t e r o f t h e s p h e r e . M

h a s t h e s a m e d i s t a n c e r t o a l l f a c e s o f

t h e p y r a m i d . T h e b i g p y r a m i d c a n b e

d i v i d e d i n t o 4 s m a l l o n e s :

M_A B C w i t h V_{1} = \frac{r}{3} A_{Δ A B C}

M_A B D w i t h V_{2} = \frac{r}{3} A_{Δ A B D}

M_A C D w i t h V_{3} = \frac{r}{3} A_{Δ A C D}

M_B C D w i t h V_{4} = \frac{r}{3} A_{Δ B C D}

V = V_{1} + V_{2} + V_{3} + V_{4} = \frac{r}{3} (A_{Δ A B C} + \dots) = \frac{r S}{3}

w i t h S = s u r f a c e o f p y r a m i d

\Rightarrow r = \frac{3 V}{S}

Commented by ajfour last updated on 24/Jul/18

w h a t i s S, s i r a n d h o w r = \frac{3 V}{S} ?

Commented by MrW3 last updated on 24/Jul/18

Commented by MrW3 last updated on 24/Jul/18

W h e n w e k n o w t h e a r e a a n d p e r i m e t e r

o f a t r i a n g l e t h e n w e c a n g e t t h e r a d i u s

o f i t s i n s c r i b e d c i r c l e :

\frac{r P}{2} = A \Rightarrow r = \frac{2 A}{P}

I u s e d t h i s m e t h o d f o r c a l c u l a t i n g

t h e r a d i u s o f i n s c r i b e d s p h e r e .

Commented by ajfour last updated on 24/Jul/18

U n d e r s t o o d S i r, t h a n k s; i t s t h e b e s t

w a y t o a r r i v e a t t h e a n s w e r .

Commented by MrW3 last updated on 24/Jul/18

I f w e s h o u l d d e t e r m i n e t h e r a d i u s o f

t h e i n s c r i b e d s p h e r e o f a t r i a n g u l a r

p y r a m i d w i t h s i d e s a, b, c, p, q, r, t h e n

t h i s m e t h o d s h o u l d b e t h e e a s i e s t o n e .

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