Question

If a rectangle and a parallelogram have same perimeter,show that area of parallelogram is less than area of rectangle

Answer 1

Answer:

Area of ||gm = base × height, And

Area of rect. = length × width,.

Suppose,W of rect. = B of ||gm

And,L of rect. = H of ||gm

So, Area of rectangle = Area of parallelogram

Here, In area, let height of parallelogram be h,

Breadth be a ,And

length be b.

So, sin(α)=ha

Then h=a sin(α) so Area of the parallelogram = (ab)sin(α)

It will form alpha in parallelogram.

Here, alpha in ||gm is less than 90∘, so sin(α) <1.

Now, we will multiply both sides of that inequality(in text. and ||gm) by ab, and we will get:

(ab)sin(α) < ab

…which means: Area of parallelogram < Area of rectangle.

Note please: sin(a),sin(ab),............ = sina, sinab ............

Answer 2

Answer:

• Here are a rectangle (black) and a parallelogram (red) that have the same base b. If the perimeters are equal, that means the sum of all sides of the parallelogram is the same as the sum of all sides of the rectangle, so, their non-horizontal sides a must also be equal.

• Do you see that the red triangular bit, hanging out of the black rectangle to its right, exactly matches the slot at the left of the red parallelogram? So you could cut the triangle hanging outside the rectangle at the right, and tuck it inside the black rectangle, at the left, to sort of “fill up the gap”; but then you’d still have a gap above the red parallelogram, under the top side of the rectangle. Intuitively, you can see that the area of the red parallelogram is less than the area of the black rectangle (you can tuck all the red area into the black one, but you’d have a gap left).

If you’ve done a bit of trigonometry, you can easily prove it:

→Area of the rectangle is base times height = a x b = ab.

→Area of the parallelogram is also base times height = b x h. To calculate its height, look at the red, right-angled triangle at the bottom-left:

sinx= h/a

Then h=asinα so

→Area of the parallelogram = absinα

Angle α (in green) is less than 90∘ , so sinα <1 .

→Now multiply both sides of that inequality by ab, and you get:

→absinα<ab

which means:

→Area of parallelogram < Area of rectangle.