Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 40469 by MrW3 last updated on 22/Jul/18

Commented by MrW3 last updated on 22/Jul/18

Find the volume of pyramid.

$${Find}\:{the}\:{volume}\:{of}\:{pyramid}. \\ $$

Commented by MJS last updated on 22/Jul/18

Euler′s formula: (1/(12))(√(a^2 p^2 f_a +b^2 q^2 f_b +c^2 r^2 f_c −δ)) f_a =−(a^2 +p^2 )+(b^2 +q^2 )+(c^2 +r^2 ) f_b =(a^2 +p^2 )−(b^2 +q^2 )+(c^2 +r^2 ) f_c =(a^2 +p^2 )+(b^2 +q^2 )−(c^2 +r^2 ) δ=a^2 b^2 c^2 +a^2 q^2 r^2 +b^2 p^2 r^2 +c^2 p^2 q^2

$$\mathrm{Euler}'\mathrm{s}\:\mathrm{formula}: \\ $$$$\frac{\mathrm{1}}{\mathrm{12}}\sqrt{{a}^{\mathrm{2}} {p}^{\mathrm{2}} {f}_{{a}} +{b}^{\mathrm{2}} {q}^{\mathrm{2}} {f}_{{b}} +{c}^{\mathrm{2}} {r}^{\mathrm{2}} {f}_{{c}} −\delta} \\ $$$${f}_{{a}} =−\left({a}^{\mathrm{2}} +{p}^{\mathrm{2}} \right)+\left({b}^{\mathrm{2}} +{q}^{\mathrm{2}} \right)+\left({c}^{\mathrm{2}} +{r}^{\mathrm{2}} \right) \\ $$$${f}_{{b}} =\left({a}^{\mathrm{2}} +{p}^{\mathrm{2}} \right)−\left({b}^{\mathrm{2}} +{q}^{\mathrm{2}} \right)+\left({c}^{\mathrm{2}} +{r}^{\mathrm{2}} \right) \\ $$$${f}_{{c}} =\left({a}^{\mathrm{2}} +{p}^{\mathrm{2}} \right)+\left({b}^{\mathrm{2}} +{q}^{\mathrm{2}} \right)−\left({c}^{\mathrm{2}} +{r}^{\mathrm{2}} \right) \\ $$$$\delta={a}^{\mathrm{2}} {b}^{\mathrm{2}} {c}^{\mathrm{2}} +{a}^{\mathrm{2}} {q}^{\mathrm{2}} {r}^{\mathrm{2}} +{b}^{\mathrm{2}} {p}^{\mathrm{2}} {r}^{\mathrm{2}} +{c}^{\mathrm{2}} {p}^{\mathrm{2}} {q}^{\mathrm{2}} \\ $$

Commented by MrW3 last updated on 22/Jul/18

thank you sir!

$${thank}\:{you}\:{sir}! \\ $$

Commented by MJS last updated on 22/Jul/18

there′s also a matrix−formula for a simplex in R^n (in R^1 the simplex is a line segment, in R^2 a triangle and in R^3 a tetrahedron, the Euler−formula for R^2 is the same as Heron′s A_(△abc) =(1/4)(√((a+b+c)(−a+b+c)(a−b+c)(a+b−c)))) but I couldn′t find this matrix...

$$\mathrm{there}'\mathrm{s}\:\mathrm{also}\:\mathrm{a}\:\mathrm{matrix}−\mathrm{formula}\:\mathrm{for}\:\mathrm{a}\:\mathrm{simplex} \\ $$$$\mathrm{in}\:\mathbb{R}^{{n}} \:\left(\mathrm{in}\:\mathbb{R}^{\mathrm{1}} \:\mathrm{the}\:\mathrm{simplex}\:\mathrm{is}\:\mathrm{a}\:\mathrm{line}\:\mathrm{segment},\:\mathrm{in}\right. \\ $$$$\mathbb{R}^{\mathrm{2}} \:\mathrm{a}\:\mathrm{triangle}\:\mathrm{and}\:\mathrm{in}\:\mathbb{R}^{\mathrm{3}} \:\mathrm{a}\:\mathrm{tetrahedron},\:\mathrm{the} \\ $$$$\mathrm{Euler}−\mathrm{formula}\:\mathrm{for}\:\mathbb{R}^{\mathrm{2}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{same}\:\mathrm{as}\:\mathrm{Heron}'\mathrm{s} \\ $$$$\left.{A}_{\bigtriangleup{abc}} =\frac{\mathrm{1}}{\mathrm{4}}\sqrt{\left({a}+{b}+{c}\right)\left(−{a}+{b}+{c}\right)\left({a}−{b}+{c}\right)\left({a}+{b}−{c}\right)}\right) \\ $$$$\mathrm{but}\:\mathrm{I}\:\mathrm{couldn}'\mathrm{t}\:\mathrm{find}\:\mathrm{this}\:\mathrm{matrix}... \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com