The area of the circle and the problem of Dido

Published on May 20, 2016

With the same perimeter, the circle is the polygon with the largest area. But how do you calculate thearea of the circle? Let's find out by studying the problem of Dido! To establish his own city, he sought to delimit the most space possible by using a rope stretched so as to form a semicircle.

Dido was a phoenician princess who fled with some followers from his native city of Tyre after he discovered that the king Pygmalion, her brother, had murdered her husband stealing the kingdom. After a long journey landed on the coast of Libya, where he asked the king Iarba, a plot of land on which to build a new city. The king promised as much land as she could contain a the skin of the ox.

Dido, without losing heart, cut the skin into thin strips and tied together to form a long rope. With this rope, the princess enclosed a part of the territory, ensuring a convenient outlet to the sea.

To enclose the greater part of the land, Dido dispose of the rope in a semicircle: but it will be the best choice? Yes! Among all the figures with the same perimeter, the circle has the maximum area! With this choice, Dido was able to delimit what would later become the territory of Carthage.

The circle is the piano part contained within a circle. To find the area of the circle of radius $$r$$ we use the formula:

$$A = \pi \cdot r^2$$

But from where we derived this formula? With a short demonstration, we found that the area of the circle is equivalent to the area of a parallelogram that has base equal to one-half of the length of the circumference and the height equal to the radius. You can see it very well with a washer of orange: the circle can be decomposed into many small triangles that, positioned correctly, form a parallelogram. The circle and the parallelogram are the two figures equiscomponibili, and then have the same area!

Curious to learn more? Look at the lesson on thearea of the circle and work out with the exercises!

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