Area of Triangles 

and other Regions

 
ObjectiveTo illustrate two methods for computing the area of a triangle (and other regions) using the coordinates of the vertices. Both methods lead to interesting extensions which can provide topics for student investigation at a variety of academic levels.

Level: High school algebra, precalculus, and matrix or linear algebra.

Prerequisites: Familiarity with general polygons, the standard formula for the area of a triangle (Area = (1/2)base*height), and the area of a trapezoid. Certain proofs use trigonometry and in particular the Law of Cosines.

Platform: No particular platform is required. However, we do cite URLs where various implementations are available for computers and calculators.

Instructor's Notes:

Triangles have long been a fascination for mathematicians and students. The pervasive triangle appears as a part of investigations in a variety of mathematical topics. Here we look at two methods for computing the area of a triangle that can be used when only the coordinates of it vertices are available. The validity of these results can be established using a combination of geometry, trigonometry, and algebra. We also cite a number of applications and extensions that can be used on a variety of academic levels as student projects. 

The area of a triangle is part of the early mathematical curriculum. The standard approach states that the area of triangle ABC is given by

 

where the height is the length of an altitude drawn from a vertex to the opposite side, called the base. See Figure 1 in which the base is segment AC. (Of course the altitude is drawn perpendicular to the base.)

Figure 1.

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 courses.

Method 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

Figure 2.

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

  http://www.math.umn.edu/~rejto/1151/1151_heron.pdf        

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 [1], [2], and [3]. 

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 

Figure 2a.

between points in the plane to obtain the lengths of the sides and then determine the semiperimeter s. 

Method 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. 

Figure 3.

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 area() 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 next section.

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 ,y3).

Application 1. 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.)

Application 2. Choose 3 positive values for a, b, and c in (1). Does the formula in (1) always make sense? Explain.

Application 3. 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 as follows:

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 appropriate conjecture. 

For calculus students modify the statement as follows:

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 statement

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

Application 4. 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.

Extension 1. 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 states, 

"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/

Extension 2. To develop a simple formula for the area formula given in (2) we use dot products as follows. Expression 

can written as the dot product of the 3-vectors 

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 

and hence

where 

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 [4] and [5].

Application 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.

A nice application involving the area of golf greens is discussed in [6]. 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 projects.

Extension 3. 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 given by

Two papers containing a development and discussion of this rather nice result are [7] and [8]. 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 below.

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 polygon.

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 extensions.

References:

1. William Dunham, "An 'Ancient/Modern' Proof of Heron's Formula", The Mathematics Teacher 78 (April 1985), pp. 258 - 259. 

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. Hill, "The Area of Polygonal Regions via Dot Products", International Journal of Mathematical Education in Science and Technology, vol. 30, no. 5, 1999, pp.765-768. 

5. David R. Hill and Bernard Kolman, Modern Matrix Algebra, Prentice Hall, Upper Saddle River, NJ, USA, 2001.

6. 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 Etterchank, "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.


Credits:  This demo was developed by 

David R. Hill
Department of Mathematics 
Temple University

and is included in Demos with Positive Impact with his permission.



DRH   9/03/2002     Last updated 5/24/2006

Visitors since 9/25/2002