Plese derive the formula for the diagonal of cube, cubiod and rectangle.
@Prakhar: Good attempt.! Here is the proper explanation.
For rectangle:
:
Consider a rectangle ABCD having length l and breadth b.
In right ΔBCD,
diag. BD2 = BC2 + CD2 (By pythagoras theorem,as each angle is of 90°)
Thus, diagonal of a rectangle is given by , where l and b are length and breadth of a rectangle respectively.
For cuboid:
Consider a cuboid ABCDWXYZ having length, breadth and height as l, b and h repectively and AX is the diagonal of the cuboid.
Consider face ABCD, having AC as the diagonal.
In ΔABC, by Pythagoras theorem,
AC2 = AB2+ BC2
⇒ AC2= l2 + b2 ... (1)
Consider right ΔACX,
diagonal AX2 = AC2 + CX2
⇒ AX2 = (l2+ b2) + h2 (Using (1))
⇒ AX2 = l2 + b2 + h2
Thus, diagonal of cuboid is given by , where l, b and h are the length, breadth and height of the cuboid respectively.
For cube:
We know diagonal of cuboid is given by . (Proved above)
For cube l = b = h = a (say)
Thus, diagonal of cube =