Question

how do i determine the geometric center of a 3D Cube?

Answer 1

Geometric conter of a 3-D cube is the point (eriside the cube) which is equidistant from all the vertices of the cube To locate this point geometrically we can use the fact that the cliagonals of a squan e or rectangle bisect each other. Also, these dicegonale are equal in length Consider a cube of edge . longth'a. The length of me face diagont of any of the 6 faces is raa. Consider the reeloogle formed by body ciągonals g two oppont fees and the relevant egcles. (See figur - rectangle ABCD) . Thaman ponathahe The diagonals of this reclongle are body diagonals of the cube. So the mid-point of these diagonal (po. O where they intersect each oner) is the geomeine conter of me cube. (Note: ' OA = 0B = OC = OD). Consider JADC O ABCD. By pymaporos theorem: AC = BD = √(iza)? + a = √2a7a² = √322 2 & AC = B. D = 3a the other forer so OA = OB = 0C = OD = 43 a. By symmetry of me cube, this point o is at the same distance (Va) from Thenfor, Geometric center of the cube is at a distonce 3 a from any restes along the body diagonal paring Through the verten.

Altemalively. Consider segment MNL pawing Through o, and parallel to sides AD and BC of the cube intersecting face diagonal AB and Co at M and N. repectively It can be easily proved geometrically mat M and N are muc-points & me face diagnols AB anal CD. and o in the mid-port of mr. So, geometric conter of the cube is at mid-point of the segment joining mid-points of the face diagonal of two opposite taw.. of the cube, * Using Coordinate geometry: Let three edge, o emanating from a restex are coincident with the coordinal . Now wing the coordinals of me nostres of me cube (see figun) we can find the equation of any of the two locy drogonats 1000 (02,0) The point of intersection of the two body diagonal (a, a,o) is the geometne cener of the cube axo), (0,a,a) 10,0,0) (ao, al 1a,0,0)