Need the Area of a Triangle? The Pope Can Help! – Explorations for Students, Part 2

Explorations for Students 2

1. Show that, for an equilateral triangle of side 30, the square of the altitude is indeed 675.
2. Show that, for an equilateral triangle of side \(b\), using Gerbert’s algorithm up through the instruction to “deduct a quarter of it” gives the correct value for the square of the altitude.
3. Suppose you are a student in Gerbert’s geometry class, using his textbook, and you have just seen his solution for how to find the area of an equilateral triangle of side 30. Now you are working on your homework assignment.
    
   • You must find the area of an equilateral triangle of side 28. What goes wrong? What number do you need to add to get a perfect square? How far off is the value for the altitude given by this algorithm? How far off is the area? Can you get a closer answer by subtracting instead of adding something?
   • What if the measure of the side had been 29? What makes this problem more complicated?