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Question Number 3843 by Rasheed Soomro last updated on 22/Dec/15

$$\mathcal{L}etthesideofthefollowingmentioned$$$$figuresis\mathbf{s}:$$$$Theareaofsquareis{\mathbf{s}}^2,the3Darea\left(volume\right)$$$$ofacubeis{\mathbf{s}}^3,the4Darea/volumeof4D$$$$hypercubecanbesaid{\mathbf{s}}^{4}andsoon.$$$$$$$$Nowifradiusis\mathbf{r},theareaofcircleis\mathit{\pi }{\mathbf{r}}^2,$$$$the3Darea\left(volume\right)ofasphereis\frac{4}{3}\mathit{\pi }{\mathbf{r}}^3,$$$$whatwillbethe4Darea/volumeof4D$$$$sphereand5Darea/volumeof5Dsphere?$$

Commented by prakash jain last updated on 22/Dec/15

$$\mathrm{Volume}{V}_n=\frac{{\pi }^{n/2}{r}^n}{\mathrm{\Gamma }(1+\frac{n}{2})}$$$$\mathrm{Surface}\mathrm{area}{S}_{n-1}=\frac{2{\pi }^{n/2}{r}^{n-1}}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}\left(surfacearea\right)$$

Commented by Rasheed Soomro last updated on 22/Dec/15

$$\mathrm{Volume}{V}_n=\frac{{\pi }^{n/2}{r}^n}{\mathrm{\Gamma }(1+\frac{n}{2})}$$$${\mathrm{V}}_3=\frac{{\pi }^{3/2}{r}^3}{\mathrm{\Gamma }(1+\frac{3}{2})}\stackrel{?}{=}\frac{4}{3}\pi r^3$$$$For4\mathrm{D}{\mathrm{S}}_{4-1}={\mathrm{S}}_3=\frac{2{\pi }^{4/2}{r}^{4-1}}{\mathrm{\Gamma }\left(\frac{4}{2}\right)}[Howmanydimentions?]$$$$\mathrm{W}econsidersurfaceas2\mathrm{D}$$

Commented by prakash jain last updated on 22/Dec/15

$$\mathrm{For}3-Dsurfacearea{r}^2$$$$\mathrm{For}4-Dsurfacearea{r}^3$$$$S_{n-1}\mathrm{is}\mathrm{surface}\mathrm{area}\mathrm{equivalent}ndimentional$$$$sphere.W\mathrm{ill}\mathrm{be}\mathrm{proportional}\mathrm{to}{r}^{n-1}.$$$$\mathrm{\Gamma }\left(\frac{5}{2}\right)=\frac{3}{2}\mathrm{\Gamma }\left(\frac{1}{2}\right)=\frac{3\sqrt{\pi }}{4}$$

Commented by Rasheed Soomro last updated on 24/Dec/15

$$\mathcal{TH}\alpha nk\mathcal{S}!$$$$\mathcal{I}fforndimentionssurfaceisofn-1$$$$dimentions.Thisisconsistantwith3\mathrm{D}$$$$figures.Becausein3\mathrm{D},surfaceis2\mathrm{D}.$$

Answered by Filup last updated on 22/Dec/15

$$\mathrm{Volume}\mathrm{is}\mathrm{area}\mathrm{rotated}\mathrm{around}\mathrm{axis}.$$$$\therefore 4Darea/volumeisthevolume$$$$rotatedaroundanaxis.\mathrm{Where}\mathrm{you}\mathrm{rotate}$$$$\mathrm{the}\mathrm{whole}3\mathrm{D}\mathrm{solid}\mathrm{around}\mathrm{the}z\mathrm{axis}$$$$\left(\mathrm{I}\mathrm{think}z\mathrm{axis}\right)$$

Commented by Filup last updated on 22/Dec/15

$$thiswasmeanttobeacomment$$

Commented by Rasheed Soomro last updated on 22/Dec/15

$$Formulaofvolumeof4Dsphere,interms$$$$ofrisrequired.$$$$Formulasofsquare,cube,hypercubeetc$$$$formaGP:{\mathbf{s}}^2,{\mathbf{s}}^3,{\mathbf{s}}^4,\dots$$$$Formulasofcircle,sphere,etcformasequence$$$$\mathit{\pi }{\mathbf{r}}^2,\frac{4}{3}\mathit{\pi }{\mathbf{r}}^3,\dots .$$$$\mathcal{W}hatisnextterm?$$$$\mathcal{I}fweaddonetermatthebegining,thefirstsequence$$$$shouldbe:\mathbf{s},{\mathbf{s}}^2,{\mathbf{s}}^3,\dots$$$$\mathbf{s}maybeconsideredlengthoflinesegment.$$$$Thesecondsequence,Ithinkshouldstartwith0$$$$\left(0canbeconsideredas{}^{\prime }{length}^{\prime }ofapoint\right):$$$$0,\mathit{\pi }{\mathbf{r}}^2,\frac{4}{3}\mathit{\pi }{\mathbf{r}}^3,\dots$$$$Point,Circle,Sphere,\dots$$

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