Need the Area of a Triangle? The Pope Can Help! – Explorations for Students, Part 1

Explorations for Students 1

1. The arithmetical rule for finding area, as we have seen, comes from the description for a triangular number, usually written \[1 + 2 + \ldots + n = \frac{n(n + 1)}{2},\] where \(n\) is a positive integer. But could the rule also be used for triangles with non-integer sides?
    
   • Use the arithmetical rule \[A = \frac{b(b + 1)}{2}\] to find the (approximate) area of an equilateral triangle of side 2. (We will omit units for the sake of simplicity.)
   • Now use it for an equilateral triangle of side 3.
   • What result does it give for an equilateral triangle of side \(2\frac{1}{2}\)? Given your previous answers, does it seem reasonable?
   • What result does it give for an equilateral triangle of side \(\sqrt{5}\)? Again, does it seem to work?

2. Exact and approximate answers
    
   • Sketch an equilateral triangle of side 7. Draw in its altitude. What is its exact value? How do you know?
   • What is the true area of an equilateral triangle of side 7?
   • You have access to two tools which Gerbert did not have: the decimal representation of a real number, and a hand-held calculator. Find a decimal approximation for the area of an equilateral triangle of side 7. Round to 3 decimal places.
   • What result did Gerbert get, using his \(\frac{6}{7}\) rule? How far off was he?
   • What about for the triangle with side 30? Find a decimal representation for the correct area and compare it to Gerbert’s result. How far off was he this time?
   It appears that things get worse, the bigger the triangle. But there is another way to look at error: the relative error is defined as \(\frac{error}{true  value}\).
    
   • For the triangle of side 7, the error was _____ and the exact area (to 3 decimal places) was _____, so the relative error was the ratio of those two numbers, or __________.
   • Find the relative error for the triangle of side 30. What do you notice?
   • In general, we might say that Gerbert’s special \(\frac{6}{7}\) rule for finding the area of an equilateral triangle has an error of about _____%. Not bad!

3. Why \(\bf{\frac{6}{7}}\)?
   All equilateral triangles are similar, so if we can find the altitude of an equilateral triangle of side 1, we can find the altitude of any such triangle, simply by multiplying the result by the side length \(b\).
    
   • Use the Pythagorean Theorem to find the (exact) measure of the altitude of an equiangular triangle of side 1.
   • Use a calculator to find decimal approximations to \(\frac{6}{7}\) and \(\frac{\sqrt{3}}{2}\), say to five decimal places. How close was Gerbert’s approximation?
   • Consider all the simple fractions between 0 and 1 of the form \(\frac{n}{n + 1}\): \[\frac{1}{2}, \frac{2}{3}, \ldots, \frac{9}{10}. \]

   Which of them is closest to \(\frac{\sqrt{3}}{2}\)? Do you think Gerbert knew that?