Expert: Vishwas K Hajirnis - 2/9/2004

Question
Why is the formula for area for kites A = (½) d1 x d2?  Where did the 1/2 come from?  

Answer
Let us draw a 'kite' first.

Let the kite have vertices ABCD with diagonals
AC and BD intersecting at point O (these will be at right
angles to each other), with side AB equal to side AD and
side BC equal to side CD. Now let the length AC be d1
and lenght BD be d2.
Kite diagonals are perpendicular to each other and BD
is bisected at O, i.e BO = OD. But BD is having lenght d2
So BO = OD = 1/2 d2

1. Now area of ABCD = Area of ABC + Area of CDA
                   = 1/2 (length AC * length BO )
                    +1/2 (length AC * length OD )
(Since area of a triangle is 1/2 * base * height,
I am using '*' to denote multiplication symbol )
                   = 1/2 ( d1 * (1/2)d2)
                    +1/2 ( d1 * (1/2)d2)
                   = 1/2 ( d1 * (1/2)d2 + d1 * (1/2)d2)
                   = 1/2 d1 * ( (1/2)d2 + (1/2)d2 )
                   = 1/2 d1 * d2

ALTERNATIVELY

2. Draw 2 lines parallel to diagonal AC, passong thru
  B and D respectively
  Similarly draw 2 lines parallel to diagonal BD
  and passing thru A and C respectively.
  These diagonals intersect each other at 4 points
  which will form a rectangle.
  Let this rectangle be called PQRS such that
  the point B is between P,Q, point C is between
  Q,R, point D is between R,S and point A is between
  S,P.
  Now the area of this rectangle is clearly
  d1 * d2, where as the kite ABCD is having area
  equal to half the area of PQRS.
  Hence you get the 1/2 term in the formula
  for area of the kite.

Hope this is clear