Cuboid and Cube
A brick, chalk box, geometry box, match box, a book, are all examples of a cuboid. A cuboid has six rectangular regions as its faces.
These are ABCD, ABFE, BCGF, EFGH, ADHE and CDHG. Out of these, opposite faces ABFE and CDHG; ABCD and EFGH and ADHE and BCGH are respectively congruent and parallel to each other.
The two adjacent faces meet in a line segment called an edge of the cuboid. For example, faces ABCD and ABFE meet in the edge AB. There are in all 12 edges of a cuboid.
Points A,B,C,D,E,F,G and H are called the corners or vertices of the cuboid. So, there are 8 corners or vertices of a cuboid.
At each vertex, three edges meet. One of these three edges is taken as the length, the second as the breadth and third is taken as the height (or thickness or depth) of the cuboid. These are usually denoted by l, b and h respectively. Thus, AB (= EF = CD = GH) is the length, AE (=BF = CG = DH) is the breadth and AD (= EH = BC = FG) is the height of the cuboid.
Three faces ABFE, AEHD and EFGH meet at the vertex E and their opposite faces DCGH, BFGC and ABCD meet at the point C. Therefore, E and C are called the opposite corners or vertices of the cuboid.
The line segment joining E and C. i.e., EC is called a diagonal of the cuboid. Similarly, the diagonals of the cuboid are AG, BH and FD. In all there are four diagonals of cuboid.
Surface Area
The surface area of the cuboid is equal to the sum of the areas of all the six rectangles or faces.
= l×b + b×h + h×l + l×b + b×h + h×l
= 2(lb + bh + hl)
Diagonal
$$ \text{Diagonal of a cuboid} = \sqrt{l^2 + b^2 + h^2} $$
Cube
A cube is a special type of cuboid in which length = breadth = height.
l = b = h
Surface area of a cube of side or edge a
$$ = 2(a \times a + a \times a + a \times a) $$
$$ = 6a^2 $$
$$ \text{Diagonal} = \sqrt{a^2 + a^2 + a^2} = \sqrt{3}a $$
Volume
Volume of a cuboid = length × breadth × height
= lbh
As cube is a special case of cuboid in which l = b = h
$$ \text{Volume of a cube of side a} = a \times a \times a = a^3 $$