Euclid’s Geometry: Quadrature of a Rectangle

A “quadrature” is a historical mathematical process to determine an area. In other words, it is constructing a square with an area equal to that of a figure like a rectangle, a triangle or a circle bounded by a curve.

Greek mathematicians were obsessed with symmetries, structures of geometric shapes and visual beauty. In the hands of ingenious ancient Greek geometers, a compass and a straightedge first turned into lines, angles, parallels, perpendiculars and then secondly they turned into regular polygons of beauty.

For instance, Euclid loved to build complex structures from simple shapes with only using a compass and a straightedge. Surprisingly we still use his constructions. And of the most challenging problem that he was dealing with was the quadrature or squaring of a plane figure.

Let’s do some example and start with a rectangle.

Let ABCD be an arbitrary yellow rectangle. We will only use a compass and a straightedge for our construction. Then we will get a square which will have the same area as our yellow rectangle.

  • First, we will extend line AB to the right, and we will get point E and segment BE. But, the length of the segment BE and the length of the segment BC are going to equal. We can say BC = BE. It is easy to have this by using a compass and a straightedge.

  • Second, we will bisect AE at center point F. Again it is easy to have this with a compass and a straightedge. Since F is the center, the segment AF and the segment FE are going to be a radius of our semicircle and AF = FE.

  • Finally, at point E, we will draw a perpendicular line to the segment AE until the point G which is the point of the intersection of the perpendicular line and the semicircle. The segment BG will help us construct our new blue square and will be the one side of our square. We will call it GHJB.

Now we can claim and then prove that the area of the rectangle ABCD and the area of the square GHJB are equal.

Proof:

Now let x, y, z be the lengths of the segments FG, FK, and GK respectively. Since GK is a perpendicular line, FGK is a right triangle. If you have a right triangle, life gives us a chance to apply the beautiful Pythagorean theorem. And it gives us;

  • x² = y² + z². And this gives us
  • z² = x² — y².

Since FK, AF, FG are radii of the semicircle, FK = BG = HG = x. And

  • BE = FE — BE = x — y, and
  • AB = AF + FB = x + y.

Now we can find the areas;

Area of the rectangle ABCD: (height) X (base) = (AB) X (BC)

(x+y) • (x — y) = x² — y²

Area of the square GHJB:

z • z = z² which is also equal to x² — y².

Thus the area of the rectangle and the area of the square are equal. Q.E.D.

Q.E.D. is usually placed at the end of a mathematical proof to indicate that the proof is complete. It means “Quod Erat Demonstrandum”.

Beautiful. We have proved that the original rectangular area equals the area of the square which we just constructed with a compass and straightedge. And this completes the rectangle’s quadrature.

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