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Question Number 5398 by FilupSmith last updated on 13/May/16

$$\mathrm{If}\mathrm{we}\mathrm{have}\mathrm{an}n-\mathrm{dimensional}\mathrm{cube}.$$$$\mathrm{How}\mathrm{do}\mathrm{we}\mathrm{find}\mathrm{its}{}^{\prime }{volume}^{\prime }?$$

Commented by Rasheed Soomro last updated on 13/May/16

$$\mathrm{Suppose}\mathrm{s}\mathrm{is}\mathrm{measure}\mathrm{of}\mathrm{side}\mathrm{of}$$$$\mathrm{n}-\mathrm{dimensional}\mathrm{cube}.$$$$\mathrm{volume}\left(\mathrm{v}\right)\mathrm{will}\mathrm{be}$$$$\mathrm{v}={\mathrm{s}}^{\mathrm{n}}$$$$.........................................$$$$0-\mathrm{dimensional}\mathrm{cube}\left(\mathrm{point}\right):$$$${}^{\prime }{\mathrm{volume}}^{\prime }(?)={\mathrm{s}}^0=1$$$$1-\mathrm{dimensional}\mathrm{cube}\left(\mathrm{line}\mathrm{segment}\right):$$$${}^{\prime }{\mathrm{volume}}^{\prime }\left(\mathrm{length}\right)=\mathrm{s}={\mathrm{s}}^1$$$$2-\mathrm{dimensional}\mathrm{cube}\left(\mathrm{square}\right):$$$${}^{\prime }{\mathrm{volume}}^{\prime }\left(\mathrm{area}\right)=\mathrm{s}.\mathrm{s}={\mathrm{s}}^2$$$$3-\mathrm{dimensional}\mathrm{cube}\left(\mathrm{cube}\right):$$$$\mathrm{volume}=\mathrm{s}.\mathrm{s}.\mathrm{s}={\mathrm{s}}^3$$$$4-\mathrm{dimensional}\mathrm{cube}\left(\mathrm{hyper}\mathrm{cube}\right):$$$${}^{\prime }{\mathrm{volume}}^{\prime }=\mathrm{s}.\mathrm{s}.\mathrm{s}.\mathrm{s}={\mathrm{s}}^4$$$$....................$$$$..............$$$$......$$$$\mathrm{n}-\mathrm{dimensional}\mathrm{cube}\left(\mathrm{hyper}\mathrm{cube}\right):$$$${}^{\prime }{\mathrm{volume}}^{\prime }=\mathrm{s}.\mathrm{s}....\mathrm{s}\left(\mathrm{n}\mathrm{factors}\right)={\mathrm{s}}^{\mathrm{n}}$$$$$$

Commented by Rasheed Soomro last updated on 13/May/16

$$\mathrm{Can}\mathrm{we}\mathrm{think}\mathrm{of}-1-\mathrm{dimensional}\mathrm{cube}$$$${}^{\prime }{\mathrm{volume}}^{\prime }={\mathrm{s}}^{-1}=\frac{1}{\mathrm{s}}$$$$\mathrm{And}\mathrm{so}\mathrm{on}.$$

Commented by prakash jain last updated on 13/May/16

$$\mathrm{There}\mathrm{was}\mathrm{an}\mathrm{earlier}\mathrm{question}\mathrm{about}\mathrm{surface}$$$$\mathrm{area}\mathrm{and}\mathrm{valume}\mathrm{of}n-\mathrm{sphere}.$$

Commented by Yozzii last updated on 14/May/16

$$Whataboutotherfiguresindimensions$$$$higherthan3,likecuboids,spheres,$$$$cylinders,torus,frustrum,etc.?$$$$Isthereanygeneraltheoremwith$$$$regardtothis?$$

Commented by Rasheed Soomro last updated on 14/May/16

$$\mathrm{The}\mathrm{anologues}\mathrm{of}\mathrm{the}\mathrm{mentioned}\mathrm{figures}$$$$\left(\mathrm{in}\mathrm{various}\mathrm{dimensional}\mathrm{spaces}/\mathrm{plane}\right)$$$$\mathrm{should}\mathrm{be}\mathrm{considered}\mathrm{indidually}.$$$$\mathrm{For}\mathrm{example}\mathrm{cuboid}\mathrm{has}\mathrm{anologues}$$$$point,line-segment,rectangle,cuboid,$$$$...in0,1,2,3....dimensionalspacesrespectively.$$$$\mathrm{If}{l}_1,l_2,l_3,...,l_{n}aredmensionalmeasuresof$$$$n-dimensionalcuboid,thenvolumewillbe$$$$l_1{l}_2{l}_3....l_n.Ofcourseanologuesofsomefigures$$$$andtheircharacteristics(volume,surface-area$$$$etc)aresomewhatdifficulttoconsider.$$$$$$

Commented by Yozzii last updated on 14/May/16

$$Interesting...$$

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