Do you know how to find the area of a quadrilateral?
What do you think of
when you think of a square? It means that all four sides are equal in shape.
But a quadrilateral is a shape whose four sides are not equal. All four sides
are of different sizes.
I hope you'll remember
that when we look at area from now on we need to pay attention to the diagonals
along with the product of adjacent sides.
When you think of a
rectangle, do you think of equal length sides, equal width sides, opposite
sides equal and parallel, and their diagonals equal?
But does it also come
to mind that the diagonals are not equal even though the opposite sides of a
parallelogram like a slanted rectangle are equal and parallel?
Not only the sides of a
quadrilateral are not equal but also the diagonals. Also, opposite sides of a
quadrilateral are neither equal nor parallel.
Perhaps if four sides
are not equal but only one pair of opposite sides are parallel, then it is a
trapezium. No pair of opposite sides of a quadrilateral is parallel. A quadrilateral
is a trapezium if any pair of opposite sides are parallel.
Why are I saying so
much you ask? Just to paint a picture in your mind of the quadrilateral.
Shall we draw a
quadrilateral now? Usually I tell such a method in the classroom so that the
students can understand it easily. I will tell you the same. If you know of a
simpler method, please share it.
Draw a tank-like
structure. Extend one side of it. Now cover the long tank side to small tank
side with a line. Is that all there is to the quadrilateral? I have illustrated
this in three steps in the figure below.
Now let's find the area
of the quadrilateral? It
can be easily found.
Draw a diagonal joining
any two corners of the quadrilateral. Didn't I say earlier that the diagonal
will play an important role in finding area? What have you drawn? That is, draw
a diagonal in the same quadrilateral we have drawn.
Now look up and down
along that diagonal. Is there a triangle above? Is the triangle below the same?
So if the area of these two triangles is added together what is the area of the quadrilateral? You
say yes? That's the point.
Now take the diagonal
of the quadrilateral by the variable d, which is the initial of diagonal. That
will be the base of our two triangles. So let's denote the variable b, the base
of the triangle, by the variable d, the diagonal of the quadrilateral. Take the
height of the upper triangle as h1 and the height of the lower
triangle as h2. I have also shown them in the picture for your
understanding.
Do we know that the
area of a triangle is ½bh square units? Now since the base of our triangles is
d, the area of the upper triangle is ½dh1 square units and the area of the lower triangle is
½dh2 square units, right?
Now area of quadrilateral = area of
triangle above + area
of triangle below right?
Accordingly,
Area of quadrilateral = ½dh1
+ ½dh2
Taking out the common
½d and bracketing it,
Area of quadrilateral = ½d( h1
+ h2 ) square units, right?
This is the formula for
finding the area of a quadrilateral. What do you understand?
Tomorrow we will find
the formula for finding the area of a rhombus in a similar way.
*****
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