Wednesday, March 25, 2009
Wednesday Math, Vol. 65: Heron's Formula
Physically, no geometric shape is simpler than a triangle. Three points in space connected by three straight lines, which must be all included on some plane in space. Triangles also have a lot of useful physical properties, most notably that they are naturally rigid constructions.
Mathematically, the simplest shapes around are rectangles. If I tell you the lengths of the sides of a rectangle, let's call them h and w for height and width, the perimeter is 2h + 2w and the area is hw. If I tell you the lengths of the sides of a triangle, call them a, b and c, the perimeter is easy, a + b + c, but what about the area? You might remember from your last geometry class all those years ago that the area of a triangle is 1/2(height x base), but I haven't told you the height, and any of the three sides can fairly be used as the base. I have drawn a triangle with side lengths 6, 5 and 5 here, and since the side of length 6 is horizontal, it's natural to think of it as the base, but it's not illegal to use any of the others as the base. The problem here is that the height is unknown, no matter what side we use as the base. Is there an easy solution?
Glad you asked, hypothetical question asker. There is a formula, but many high school geometry classes do not include it, because it's nowhere near as "easy" a formula as 1/2 (height x base). This is Heron's Formula, named for Heron of Alexandria who published his work in 60 A.D. The symbol s stands for the semi-perimeter, s = 1/2(a + b + c). While it looks messy and it is possible the answer will be a square root, and therefore not a rational number, let's plug in the numbers we have from the triangle at the top to see how it works.
p = 5 + 5 + 6 = 16
s = 16/2 = 8
Area = sqrt[8(8 - 5)(8 - 5)(8 - 6)] = sqrt [ 8 x 3 x 3 x 2] = sqrt  = 12.
This triangle has a nice whole number value for its area. That's because the height is 4 and this isosceles triangle is actually two right triangles with sides 3, 4, and 5 glued together. In a right triangle, the two short sides can be used as the height and base. Other triangles, even isosceles triangles, won't necessarily have numbers that work out so cleanly.
If you click on the link to Heron's Formula at Wikipedia, you can see the proof of why the formula works. The proof is definitely not as easy as the proof of 1/2(height x base), and the formula is more work than plugging into 1/2(height x base), but it's not that much work, and it gives us options for understanding triangles that are defined by the side lengths.