This formula is easy to use provided we
have the measurements for the length of both the base and the height.
However, in a
variety of situations we do not have this information, nor is it immediately available. We next present two methods for the computing the area of a
triangle which rely only
on the coordinates of
the vertices of the triangle. The first method appears occasionally in high
school curriculums and sometimes in precalculus or finite mathematics courses.
The second method may appear in finite mathematics courses or linear algebra
1. If further information about the angles of triangle ABC is known then we can use trigonometry to determine the height. For
the triangle in Figure 2 we have labeled the lengths of the sides with the
lower case letter corresponding to the angle opposite the side. Furthermore
suppose we know
that angle C is equal to , then using trigonometry we have that the
height is so we have
Using the Law of Cosines and algebra
involving the difference of two squares the preceding formula can be shown to
be equivalent to the following expression which is known as Heron's
Formula for the area of a triangle with sides of lengths, a, b, and c.
The quantity s
in Heron's formula is called the semi- perimeter of the triangle. (Note: The
development of the details to this approach to
Heron's formula can be used as a project for a capable precalculus student. A
nice example of this appears at
This pdf file can be downloaded either from
the previous web address or by clicking here.
Included with permission.)
Heron's formula was proved by the Greek
Mathematician Heron (ca. A. D. 75) in his Metrica. There are a variety
of proofs of Heron's formula some based on geometric arguments. See , ,
Given the coordinates of the vertices A, B,
and C as shown in Figure 2a we can compute the lengths of the sides a, b, and c and then apply
Heron's formula to compute the area of the triangle
ABC. We use the formula for the distance
between points in the plane to
obtain the lengths of the sides and then determine the semiperimeter s.
2. An alternative to Heron's formula uses a linear combination
of areas of trapezoids (area of a trapezoid = (1/2)times "its
height" times "the sum of the lengths of its parallel sides")
and produces a simple algebraic expression for the area of a triangle. For the
development of this result we start with triangle PQR in
the first quadrant. (The location is for convenience and the final result is
independent of the location of the triangle.) In Figure 3 we draw
perpendicular segments from each vertex to the horizontal axis.
We focus on the three trapezoids APQB, BQRC, and APRC and it follows that
area() = area(APQB) + area(BQRC) - area(APRC)
Computing the areas of trapezoids
APQB, BQRC, and APRC we have
Substituting these results into the expression for
and collecting terms we have
To make this expression independent of the
order in which the vertices of the triangle are labeled and the location of the
triangle we take the absolute value of the right side of the preceding
expression and get
for any triangle with vertices (x1,
y1), (x2, y2), and (x3, y3).
This expression for the area of a triangle
requires no intermediate calculations such
as the length of sides that is needed by
Heron's formula. This expression may appear a bit cumbersome, but an
easy expression using dot products is developed in
Extension 2 in the
Extensions and Applications
For ease of reference label the two formulas as follows for a triangle T with vertices given by the ordered pairs
(x1,y1), (x2 ,y2), and (x3
Use (1) to determine a formula for the height of triangle T. (Hint: See Figure
1 and the standard formula for the area of a triangle.)
Choose 3 positive values for a, b, and c in (1). Does the formula in (1)
always make sense? Explain.
Show that the maximum area of triangle T with a fixed perimeter occurs when a
= b = c. That is, T is an equilateral triangle.
Comment: This statement can be adapted for
students of various backgrounds.
For precalculus students modify the statement
Triangle T is isosceles with perimeter 100.
Suppose b and c are integers with b = c. Construct a table of values for a, b, c, and the area to form a
conjecture for the size of the third side of length a.
Using a calculator or spreadsheet and integer values of the
equal sides student can readily perform a numerical experiment to form an
For calculus students modify the statement as
Triangle T is isosceles with fixed perimeter.
Suppose b = c, set up and solve a max-min problem to form a conjecture for the
relationship between the lengths of the three sides a, b, and c.
Next upgrade the problem to the original
For students with some experience with
geometric means and in particular the inequality that says the geometric
mean is always less than or equal to the arithmetic mean (or a average), use
the original statement and possibly supply a hint about this inequality.
For additional extensions and
applications see James W. Wilson's (University of Georgia) material at http://jwilson.coe.uga.edu/emt725/Heron/Heron.html
For a very nice graphical implementation in Java of Heron's formula go to James
P. Dildine's, "Triangle Explorations" at http://www.mste.uiuc.edu/dildine/heron/triarea.html
In addition to the Java applet you will
find an Excel file, a Sketchpad file, and a TI-83 Calculator file.
A multistage project involving Heron's formula (and a number of associated
concepts) is available as a Mathwright Microworld. This interactive Web Book,
by James E. White, is part of project WELCOME and is included in JOMA
(Journal of Online Mathematics and Applications). To access the Web Book
through JOMA link to the beginning of the
JOMA article at
http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=473 and then choose the Heron link.
The introduction to this work in JOMA
"This Interactive Web Book explores the geometry behind Heron's formula for the area of a triangle in terms of its sides. The
formula may be understood by asking which quadrilateral with assigned side lengths has the largest area. This book has several
experiments embedded in its pages, one of which allows the reader to vary the shape of the quadrilaterals to discover the surprising
answer, and thereby, to discover Heron's formula.
Nearly all of the topics discussed will be accessible to a student who is comfortable with Algebra and Geometry. The crucial step
of the argument, however, uses elementary Calculus and may, as a surprising application of the ideas of Limit and Derivative, be
taken as a motivation for studying those concepts."
Students will need
calculus; the Method of Lagrange Multipliers is used, but it is not required.
More Web Books, information about Project
Welcome, and Mathwright can be found at http://www.mathwright.com/
To develop a simple form
ula for the area formula given in (2) we use dot
products as follows. Expression
can written as the dot product of the
so we have
The formula in (2) can also be expressed
using matrix multiplication as follows. The vector of differences of the
y-coordinates is a matrix product involving a fixed matrix C and the vector of
y-coordinates; we have
Thus (2) has been reformulated into a
simple matrix product. This formulation makes it easy to develop results about
the magnification factor of matrix transformations in R2 and
provides a geometric foundation for determinants. (See  and .)
5. Both (1) and (2) can be used to determine the area of a closed
polygon by subdividing it into a set of non-overlapping triangles.
This process is called triangulation
and is widely used to determine areas of irregular regions. The process can
also be adapted for regions with curved boundaries so that quite accurate area
computations can be made.
nice application involving the area of golf greens is discussed in .
The paper also
discusses two other methods which use sectors of circles. These methods are
compared and analyzed. The ideas developed in this paper can be used to
illustrate "real life" problems and can be developed into student
The area of a polygon can be computed directly without triangulation given the
coordinates of its vertices. One development of the result generalizes the
procedure used to obtain the formula for the area of a triangle given in (2),
namely combinations of areas of trapezoids. Another development uses the areas
of right triangles and trapezoids. State the result next:
Label the vertices of the a polygon P1
= (x1, y1) through Pn = (xn,
yn) either in a clockwise manner of counter-clockwise
manner. Let Pn+1 = P1, then the area of the polygon is
Two papers containing a development and
discussion of this rather nice result are  and . It is noted in both papers that (3) can be
derived from Stokes theorem applied to a standard line integral.
The area of a polygon as given in (3) can
be 'discovered' by students with a bit of guidance. One such guided discovery
was used near the beginning of a linear algebra course to show a practical use
for a matrix as a data structure, lay a foundation for
area computations that would subsequently be discussed, introduce the determinant of a 2 by 2 matrix long before a formal definition of
determinants, and to provide a glimpse of ties between geometry and linear
algebra. The worksheets for this activity can be downloaded by clicking here.
(This material was contributed by David R. Hill, Temple University.) You will not find a mention of determinants in the
material since it was part of the follow up discussion after the student's work was
corrected and returned. A brief discussion of one way introduce the connection
between the triangle/polygon activity and the linear algebra topics is described
To provide an
efficient and easy to use way to compute the
expression in (3) define the 2 by (n+1) matrix
Imagine a box two columns wide which
slides along this matrix, stopping at each column to compute the products
indicated in the next diagram. We record these products, compute their sum,
take the absolute value, and then multiply by 1/2 to get the area of the
The use of linear algebra provides a
systematic way to compute (3) which has a nice visual component.
Extension 4. Comap
(http://www.comap.com) has a FIAM
(Faculty Advancement in Mathematics) module entitled Area of a Polygon which
is intended for teacher training and/or classroom use. This module contains
sections on area using determinants, Pick's formula, Lawrence's formula, and
1. William Dunham, "An 'Ancient/Modern' Proof of Heron's
Formula", The Mathematics Teacher 78 (April 1985), pp. 258
2. Bernard Oliver, "Heron's Remarkable Triangle Area
Formula", The Mathematics Teacher, 86 (Feb. 1993), pp.161 - 163.
3. Roger B. Nelson, "Heron's Formula
via Proofs Without Words", The College Mathematics Journal, 32
number 4, September 2001, pp. 290 - 291.
4. David R.
Area of Polygonal Regions via Dot Products", International Journal of
Mathematical Education in Science and Technology, vol. 30, no. 5, 1999,
David R. Hill and Bernard Kolman, Modern Matrix Algebra, Prentice
W. Gary Martin and Joao Ponte, "Measuring the Area of Golf Greens and Other Irregular Regions", The Mathematics Teacher 78
(May 1985), pp. 385 - 389.
7. Rene Stolk and George
"Calculating the Area of an Irregular Shape", BYTE, Feb.
1987, pp. 135 - 136.
8. Will Watkins and Monty Taylor,
"Calculating areas of Irregular Polygons", PRIMUS, Vol. III,
Number 4, 1993, pp. 379 - 388.