Problem Solving with Heron's Formula

PROBLEM SOLVING WITH HERON'S FORMULA

Jim Wilson
University of Georgia

Introduction

Heron's formula for the area of a triangle with sides of length a, b, c is

where

I hope to show some examples of interesting problems for which using Heron's formula might be useful. By "interesting" of course I am saying they are interesting for me. I hope they also do the trick of others.

Explorations and Problems with Heron's Formula

Exploration

For instance, one investigation could be to have students measure the sides and an altitude on several triangles and, with calculator, compute the areas with both formulas

Comparison of the results could well lead to intuitions about the areas of triangles and understanding of when one formula would be more applicable than the other. This is very open-ended but the idea is to use uncovered ideas to formulate additional inquiries.

Problem

Show that the maximum area of a triangular region with a fixed perimeter occurs for an equilateral triangle.

Comment. I would adapt the statement and context of this problem depending on the background of the students. Above, I mentioned an exploratory investigation where students looked at different triangular regions that could be formed with a perimeter of 100 feet. Now extend this. Have students organize a table where the lengths of the sides are varied systematically .

In order to vary something "systematically" one needs to identify a variable that can be ordered in the table. For example, investigate the more manageable problem of isosceles triangles. Let the side of length a be the base to vary from 2 to 48 in steps of 2, as follows
This table can be generated with a calculator in only a few minutes and the data open up lots of opportunity for discussion, plausible reasoning, and problem posing.

For example, four of the lines in the table will show triangles with integer areas. Are there other triangles (non-isosceles) with perimeter of 100 having integer sides and integer area?

Once the table is complete, consider having the students plot a graph with the length of a on the x-axis and the area on the y-axis. The resulting curve provides more opportunities for plausible reasoning.

PROOF

Let us return now to a proof that the equilateral triangle has the most area for any fixed perimeter. We will use Heron's formula and the Arithmetic Mean - Geometric Mean Inequality

and equality occurs when s - a = s - b = s - c, that is, a = b = c. Since the product is always less than this constant, this constant is the maximum for the product and the maximum area of a triangle with a fixed perimeter 2s is

The equilateral triangle has that maximum area.

Problem:

Interpret the condition

s(s - c) = (s - a)(s - b)

What can be concluded for any triangle for which the condition holds?

Problem

Which of these triangles has the largest area?

Answer via Heron:

Investigations:

Find triangles having integer area and integer sides.

Find a triangle with perimeter 12 having integer area and integer sides.

Find a triangle having integer sides and integer area that is not a right triangle. Can you find others? Generalize.

Find the smallest perimeter for which there are two different triangles with integer sides and integer area.

Find all the triangles with perimeter of 84 having integer sides and integer area. There are more than 5.