EXPLORATIONS AND PROBLEM SOLVING WITH HERON'S FORMULA

AND ALTERNATIVE PROOFS


by

Jim Wilson
University of Georgia

Presentation to NCTM Annual Meeting 17 Apr 2015, Boston.

Introduction

Heron's formula for the area of a triangle with sides of

length a, b, c  and the semiperimeter is

A Modern (and Elegant) Geometric Derivation(1)

This derivation uses the incircle and the excircle to one side to determine two pairs of similar triangles that allow derivation of  the radius  r   of the incircle in terms of the lengths of the sides.

A Modern (and Elegant) Geometric Derivation(2)

Same derivation but different write-up.

A Derivation using Complex Numbers

An Algebraic Derivation

This  proof is straightforward and tedious.    It has enough chances to mess up the algebra notation but is generally easy to follow.      The altitude   h   to one of the sides is expressed in terms of the lengths of the sides and  the semiperimeter is used only at the end to simplify the formula. Alternate forms of Heron's formula are sometimes given without use of the semiperimeter.

Heron's Geometric Derivation

In this approach Heron uses some seemingly unrelated similarities and cyclic quadrilaterals to determine a value for the radius   r   of the incircle in terms of the lengths of the sides

GSP file

Using the Sine and Cosine functions

This approach is very similar to the Algebraic approach shown earlier.  It really makes minimal use of trigonometry.   In this approach, the starting point is the formula for the area of a triangle is half the product of the sine of an angle and the lengths of its adjacent sides.

 

 

 

Using the Cotangent function

In this approach I will use some notation that has been seen previously.   We have the incircle and its points of tangency with equal distances from the vertices.

The shaded triangle is a right triangle with legs of  r   and  s - a. The cotangent can readily be expressed in terms of   r  and  s - a.

 

 

 

 

Explorations and Problems

In this section, a few explorations and problems are presented, some with comments, some just stated.

Perfect Triangles

Explorations and Background

An Analysis/Solution using Heron's Formula

 

 


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