Maximizing Areas and a Formula of Heron
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This Mathwright Microworld develops Heron's formula for calculating the areas of triangles in a surprising way. It shows that Heron's classical formula is actually a special case of a more general construction on quadrilaterals.
Hero of Alexandria (ad 20?-after 62), was a Greek mathematician and scientist. His name is also spelled Heron. He appears to have been of Egyptian birth, to have done his work in Alexandria, Egypt, and to have written at least 13 works on mechanics, mathematics, and physics. He developed various mechanical devices, including the aelopile, a rotary steam engine; Hero's fountain, a pneumatic apparatus in which a vertical jet of water is produced and sustained by air pressure; and the dioptra, a primitive theodolite, a surveying instrument. He is best known, however, as a mathematician. In geometry and geodesy he handled problems of mensuration more successfully than anyone of his time. He also devised a method of approximating the square roots and cube roots of numbers that are not perfect squares or cubes. The formula attributed to him, however, for finding the area of a triangle in terms of its sides, was devised before his time.
- From "Hero of Alexandria," Microsoft (R) Encarta, 1993.
Everyone knows how to calculate the area of a right triangle in terms of the lengths of its sides. In the right triangle below with legs "a" and "b" and with hypotenuse "h" the area is just half the product of a with b:
This is because the triangle is "half" the rectangle, and the area of the rectangle is taken to be: .
But if the triangle is not a right triangle, we usually have to resort to the observation that the area is one-half the product of the length of the base with the altitude from that base, and then to use some method (for example, trigonometry) to discern the length of that "altitude". So, in the picture below, we would have to figure out what the length of "altitude" was before calculating the area:
Except in very special circumstances, this altitude is not readily expressible in terms of the side lengths: a, b, c.
Now there is a formula, attributed to Heron of Alexandria, that enables us to calculate the area of a triangle directly in terms of the lengths: a, b, and c. The formula, called Heron's Formula, is the topic of this microworld. We will see that it has a more general application than to the calculation of areas of triangles, and that it can be derived by maximizing the areas of certain quadrilaterals. Finally, we will see that these considerations lead to another rather more mysterious formula for the area of a triangle.
In the course of this exploration, you will be able to experiment with your own examples to see what is going on.
Now we describe Heron's Formula. Suppose you are given an arbitrary triangle (right triangle or not) with sides: a, b, and c, as in the picture above. The perimeter of this triangle is, of course, .
Define the semi-perimeter (semi means half) to be
We shall call the semi-perimeter of the triangle: s.
Exercise: Show that the positive numbers: a,b, and c can define a triangle only if the following conditions are satisfied:
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(Hint: The triangle inequality)
Now Heron's formula for the area of the triangle is:
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The experiment on the next page will allow you to test this formula for yourself.
