Archimedes and the measure of the circle | Science


The well-known formula of the length of the circumference, 2πr, is actually a tautology, since π is, by definition, the ratio between the circumference and its diameter (or what is the same, 2r, twice the radius). But the no less known formula of the area of ​​the circle, πrtwoIt is not obvious, and to find it, it took all the ingenuity of one of the greatest mathematicians of all time.

In his book About the measure of the circle, one of the most important scientific texts of antiquity, whose influence lasted for centuries (despite its brevity and not being kept complete), Archimedes, anticipating in 2,000 years the "indivisibles" of Cavalieri and the infinitesimal calculus of Leibniz and Newton, he deduces the formula of the area of ​​the circle while finding an incredibly accurate value of π. But, let us start at the beginning…

The only geometric figure whose formula of the area is evident, is the rectangle, since we only have to multiply the length of the base by that of the height to find the number of square units. For example, if we have a rectangle of 5 centimeters of base and 3 of height, it is evident that it will contain 5 x 3 = 15 squares of 1 centimeter on each side, that is, 15 square centimeters. Generalizing, the surface (S) of a rectangle of base b and height a will be S = b.a.

Archimedes and the measure of the circle

Archimedes and the measure of the circle

It is easy to see that any parallelogram can be converted into a rectangle by "cutting" a right triangle from one end and "gluing" it into the other, so in this case too the area will be obtained by multiplying the base by the height: S = b.a.

Archimedes and the measure of the circle

And since any triangle can be considered half of a parallelogram of equal base and height (which can be obtained by tracing two of the vertices parallel paths to the opposite sides, as shown in the figure), the area of ​​a triangle will be bh / 2 ( the height is usually designated indistinctly with the letters aoh).

Archimedes and the measure of the circle

In the case of a regular polygon, like the hexagon of the figure, which has all its sides and all its equal angles, we can divide it, by tracing its radii, in as many equal isosceles triangles (which in the case of the hexagon will be equilateral) as sides have. Therefore, your area will be n.l.a / 2, I feel n the number of sides, l the side of the polygon and at the height of each triangle, which is the apothem of the polygon; but n.l is the perimeter (p) of the polygon, then its area will be p.a / 2.

To find the value of π, Archimedes imagined a circle enclosed between an inscribed and a circumscribed polygon of an increasing number of sides. Obviously, the length of the circumference had to be always greater than the perimeter of the inscribed polygon and smaller than the perimeter of the circumscribed polygon, and from 96-sided polygons respectively inscribed and circumscribed, it found a value of π between the fractions 223 / 71 and 22/7; the average of these two values ​​is approximately 3.1418, which means that in the value found by Archimedes the error is only two ten thousandths.

And although the reasoning by which the great Greek mathematician arrives at the formula of the area of ​​the circle is somewhat longer and elaborate, it ultimately amounts to considering that the circle is a regular polygon of infinitely small infinitely small sides, so its apothem is the radius of the circle and its perimeter the length of the circumference, so that the formula pa / 2 becomes 2πr.r / 2 = πrtwo.

It is curious the way in which Archimedes presents the area of ​​the circle, saying that it is equal to that of a right triangle whose legs are, respectively, the radius of the circle and the length of its circumference. A nod to the old problem of squaring the circle, although surely Archimedes already knew that it was unsolvable. But that's another question ...

Master forms is a section by Carlo Frabetti dedicated to explaining the main formulas of mathematics and physics, their origin, evolution and precise meaning.

Carlo Frabetti He is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 scientific dissemination works for adults, children and young people, among them Damn physics, Damn mathematics or The big game. He was a screenwriter The Cristal ball.

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