# Area of Square $\neq$ the area of Rhombus created by stretched square?

I have a problem where I am given a square with each side = to 3, that is stretched until points A and B are at a distance of 3 away from each other. Additionally, the length of the sides do not "stretch".

I thought that because area is base * height, that the area would not change due to this transformation.

In this case, the area of the square before transformation is 9, but the area of the rhombus is 7.794.

Why does the area change? Why is the area of the rhombus still not base times height?

Please note, I understand how to get the area of the rhombus - it just does not make sense intuitively and conflicts with the area of a parallelogram being b times h idea. • Do the edge lengths stay the same, as the square is "stretched"? – pjs36 Nov 22 '15 at 16:53
• The lengths stay the same, as it is supposed to be a square made out of wire. Should have made that clear. – Ulad Kasach Nov 22 '15 at 16:54
• "I thought that because area is base * height, that the area would not change due to this transformation." But the height does change (and the base stays the same). – fleablood Nov 22 '15 at 17:07

It IS base times height but the height has changed.

The height is the perpendicular from line to line. As a square the height was 3 as the sides were perpendicular. You've squished the square over while keeping the sides the same but the sides are no longer perpendicular. So the height is no longer the same thing as the sides. The height is now less than 3.

• Ohhhhhh. The height is perpendicular to the base. How simple. Thank you. – Ulad Kasach Nov 22 '15 at 17:11
• Yes, The Imagine either a) a house with a lean to. The height of the house is the straight wall going perpendicularly up and down. The length of the lean to has nothing really to do with the height as it doesn't really go straight up or down or b) a shoddily made rectangular book case with weak corners so that the sides begin to lean over and fall down; the lengths stay the same but they lean over and fall to the floor. c) a ladder leaning against a wall. How high the ladder depends on the angle. – fleablood Nov 22 '15 at 20:47

The base length will not change, but the height will change. The height will no longer be a side length of the rhombus/parallelogram; it will be shorter.

The area of a rhombus and a square are not supposed to be same even with perfect stretching of square to form a rhombus.

A square is a SPECIAL RHOMBUS by all properties while a rhombus is not a SQUARE by some properties.

Two diagonals of square are the same while that rhombus are not.

Square has all its 90° while rhombus has only 2 pairs of opposite angles same.

Area of Square = s² = p²/2 (if s is the side and p is diagonal)

Area of Rhombus = pq/2 (where p and q are diagonals of the rhombus. ≠ s²

To prove this,

The diagonal of a rhombus p and q intersect at right angled and form two equal isosceles triangles.

So area of rhombus is

2 × Area of one of the triangles

But

Area of the triangle = 1/2 × base × height

                               = 1/2 × s × p/2

= ps/4 ≠ s