Notes — Chapter 9: Area

To the Student. An Olympic swimming pool is 50 meters long and 21 meters wide. Placing a meter-square grid on this rectangular pool, you can see that its area is 50 x 21 = 1050 square meters. For rectangles in general, Area = Base x Height, using consistent units. We make this fact our Postulate 9. To work out the areas of other shapes we use Postulate 8, which is a consistency principle. Area Postulate: Every polygonal region has

a number called its area such that: 1) congruent triangles have the same area, and 2) the area of a polygonal region is equal to the sum of the areas of its nonoverlapping parts.

A rectangle with base b and height hf can be split into two congruent right triangles each with the same base and height. Postulate 8 lets you conclude that the area each of these right triangles must be +Base x Height = sbh. This formula works for all triangles, once you understand of how the height, h, or altitude corresponding to the base, b, is measured. When you see an area formula, relate it to simpler formulas that you know. Linking algebra to geometry this way will let you work out areas whenever you need to.

Areas are practical, allowing you to figure out how much paint or wallpaper is needed for a room. They also are the key to proving essential geometric facts such as the Pythagorean Theorem in this chapter or the Side- Splitter Theorem in the chapter on similarity. “

ll’ About Area. This chapter builds on three facts: The area of a rectangle is the product of its length by its width. Congruent regions have the same area. And the principle that the area of a region should be the sum of the areas of its nonoverlapping parts (consistency of areas). Given these, students can find the formulas for the areas of common polygonal shapes.

This material depends on linking the geometry of the diagrams to the algebra of the formulas. Each aspect reinforces the other and strengthens understanding.

This chapter draws on and strengthens the student’s intuitive sense of geometric areas. Postulates 8 and 9 formalize these intuitions, allowing us prove the Pythagorean theorem using the area argument informally outlined in Lesson 6 of Chapter 1. Area is also used to establish the Side-Splitter Theorem in the next chapter.

Working in the Cartesian coordinate plane, it is possible to define an area function that satisfies Postulates 8 and 9. I outline this approach in a note at the end of my notes on this chapter’s lessons. This approach starts from a deep understanding of coordinate geometry and works backward to justify the intuitive assumptions used in elementary geometry. It was developed by mathematicians who knew their Euclid and wanted a deeper understanding of area. For our purposes, the understanding given by Postulates 8 and 9 is sufficient and easier to grasp.

Appropriate Technology. The tools we have for calculating areas allow a wide range for the use of algebra. Many exercises involve passing back and forth between geometry and algebra. See, for instance, the Trapezoidal Rule (37-40), Lesson 4. These and other exercises can be worked using programmable calculators or

computers, although Jacobs has not made this a requirement. Many exercises invite use of mental calculation or of a scientific calculator. See Exercises 15-30 of Lesson 5, for instance. The text does not suggest a particular approach. Teachers can suggest the means they think appropriate or to leave this as a matter of student choice.

Lesson 1. Area. The areas of various states serve as reminders of our common ideas of area. We define a polygonal region to be a polygon and its interior. We will work out the areas of such regions using The Area Postulate. see above or Geometry. Part 2 of this postulate says that the areas of the pieces add up to give to give the area of the whole 9consistency). Part 1 is a special case of the more general principle that congruent regions have the same areas (again, consistency).

The exercises show how far the simple and intuitive Area Postulate can carry students. Set I. The early examples conceptual (1-6), then come those using algebraic notation (7-19). Contrast Exercises 1-15 with Exercises 16-24. Set II. Note Exercises 46-55, Comparing Parallelograms, which anticipate later results on areas. A dissection paradox from Jerry Slocum and Jack Botermans gives us our Set III exercise.

Lesson 2. Squares and Rectangles. The basic insight comes from laying out a square grid, illustrated by the regular pattern of western townships. This insight is the basis for the theory of area sketched in my note at the end of these lesson-by-lesson comments. This insight provides the key for our work on area, Postulate 9. Area of a Rectangle: The area of a rectangle is the product of its base and altitude. In symbols, A = bh. More commonly, “Area = Length x Width,” but Jacobs uses “base (= b)” and “altitude (= h)” because these notations and ideas will carry through for the areas of triangles, parallelograms, and trapezoids.

Notice the importance of units in Set I Exercises 1-−4 and 7-10. Notice also the use of algebra here. See Distributive Property (11-17). Notice the role of calculators and calculating in Two Squares (22-25). Jacobs shows us various units used by people in different fields. Students should learn to work with whatever units are required. There is a natural evolution of units, as older systems defined in easy practical units are replaced by systems defined more generally and abstractly. For instance, the foot and inch, related to the human frame, are now defined in metric or S.J. units. The international inch is 2.54 centimeters, exactly. The meter, intended to be (1/10 000 000)th of the distance from the North Pole to the Equator, is now defined as the distance light travels in a vacuum in (1/299 792 458)th of a second. Practical measurement may use rulers, meter sticks, or paces. Students must be at ease working with any of these.

Set III. Dividing into Squares, this problem of “Squaring the Square” was solved by a group of students

at Trinity College, Cambridge. One of them, Willian T. Tutte, wrote a delightful account of their work for Martin Gardner’s Mathematical Games column in SCIENTIFIC AMERICAN. This is reprinted in Gardner’s Second SCIENTIFIC AMERICAN Book of Mathematical Puzzles & Diversions, 1961, with updates to be found in Gardner’s Fractal Music, Hypercards, and More, 1992. A charming aspect of this work is that a crucial contribution was made by Mrs. Brooks, the mother of one of the four students working on this problem. She found an unexpected solution to a puzzle her son made out of a pattern for squaring a rectangle. This discovery changed the whole direction of their effort. An exposition of the methods used can be found in Sherman Stein’s Mathematics: The Man-made Universe, available from Dover Publications in 2003. Students could use these sources for reports or projects.

Lesson 3. Triangles. The Rhind papyrus, which was written about 1650 Bc by the Egyptian scribe Ahmose, gives instructions for computing the area of a triangle. Interpreted algebraically these amount to aAABC = $ba. This is correct for a right triangle with legs of length a and b. A diagram shows that the required area is exactly half of that of a rectangle of base b and height or altitude a, so this formula follows from Postulates 8 and 9. Further diagrams show that aAABC = $bh, where b is any base of AABC and h is the corresponding altitude. The formula is part of this lesson, which includes learning to visualize this result using diagrams, express it in words and algebraically, and to use it to solve practical problems.

Set I. Geometric reasoning suffices to work out some of the exercises, for instance Math Puzzle (1-3). Algebra plays a role in others; see Isosceles Triangle (4-9). Enlargement (15-23) leads students to discover the scaling properties of perimeters and areas. We will return to scaling in Lesson 6 of Chapter 10 and in the similar solids lesson in Chapter 15.

Set II. Surfer Puzzle (29-34) returns us to the problem that opens the book. Now we see how areas can be used to show why every location on the island gives the same sum of distances for the Surfer. The area argument used here also establishes the constancy for the analogous sum for any convex, equilateral n-gon. In particular, it proves the constancy of such sums for a regular pentagon (Exercise 28, Lesson 1, of Chapter 6). For the equilateral triangle, the constancy can be proved using symmetry rather than area. You might challenge your students to find a proof for equilateral triangles that does not use areas.

Dividing The Cake (47-49) has a neat solution that students are asked to extend in Exercise 50. The Shingle Roof (51-54) shows a common practical use of geometry.

Set III asks students to explore Heron’s Theorem for finding the area of a triangle using the lengths of its

sides. The algebraic proof of this formula is routine and gives no geometric insight. Jocobs omitted it, although it is found in earlier editions. Some students have access to CAS (Computer Algebra Systems) and for them setting up and checking Heron’s formula could _ be instructive, especially seen in the light of working this out by hand, which is a project just within the capabilities of skilled and determined students. We better appreciate the value of aids such as CAS when we see how easily they deal with what is possible but difficult for us to do unaided.

Lesson 4. Parallelograms and Trapezoids. An example from D’Arcy Thompson’s classic book On Growth and Form, shows how a mathematical transformation can change the outline of a fish of one species into that of another. This transformation turns rectangles into parallelograms. In this lesson we learn how to find the areas of parallelograms and trapezoids. Simple diagrams and Postulates 8 and 9 show us that for parallelograms aABCD = hb, where b is any base of the parallelogram and h is the corresponding height or altitude. For trapezoids, we must ust the lengths of the two parallel bases, here a and b, and the height, h, is the distance between the parallel lines that these bases lie on. The area of a trapezoid is then aABCD = $h(a+0). As always, students need to understand these results visually (from the diagrams), in words, and algebraically.

Set I. From Above (1-5) is conceptual and visual. Tax Assessor Formula (6-16) returns students to the area formula of Lesson 5 in Chapter 1. In these exercises, students must see how to compute areas using the methods we have developed and compare these correct areas with the results of the Egyptian formula. This gives practice in recognizing areas that can be computed and in using formulas to compute these areas. There two themes run through the exercises of this lesson. See also the practical surveying problem of Set IT, Inaccessible Field (30-36), and the problem of estimating the area under a curve in Trapezoidal Rule (37-42). This last group of exercises ties into ideas used in calculus. For students having access to programmable calculators or Computer Algebra Systems (CAS), the Trapezoidal Rule used here could be programmed to run automatically.

▶ The Egyptian Assessor’s Estimate. This formula is a puzzle, so giving out the following solution would spoil the fun. But it is interesting that there is a geometric and algebraic argument that shows that the assessor never loses income by using his formula. The assessor’s formula comes down to estimating the area of a quadrilateral using 4 (ab + be + cd + da), where a, b, c, and d are the lengths of its sides, taken in cyclic order. Assume the quadrilateral is convex, since if it were concave we could turn it into a convex quadrilateral by popping one pair of sides out and this would increase its area. The sides a and b determine a triangle

whose area is no larger than sab. Likewise, the triangle determined by sides c and d has an area no larger than $cd. These triangles cover the quadrilateral, so Area of Quadrilateral < $(ab + cd). A similar argument shows that Area of Quadrilateral < (bc + da). Hence 2x Area of quadrilateral < $(ab+bc+cd + da). Dividing by 2 gives Area of Quadrilateral < $(ab+be+ cd + da), so the Egyptian assessor’s formula never underestimates the area. Moreover, this reasoning shows that his formula gives the “right” area only when the quadrilateral is a rectangle, and each of triangles we considered is a right triangle.

Lesson 5. The Pythagorean Theorem. The square on the hypotenuse of a right triangle is equal to the sum of the squares on its legs. ‘This is the most famous result in geometry. Its algebraic form, a? +b? = c?, is familiar to many. The linking of algebra and geometry is what allows us to calculate distances and what makes this result so useful. If a triangle’s sides satisfy a” + 6? = c*, it must be a right triangle, but this converse of the Pythagorean theorem needs a separate proof.

Set I. See Theorem 43 (4-14). The links between calculations and geometry are reinforced in Squares on the Sides (15-30). The proof using areas is intuitively appealing. This key property of right triangles has been studied down the ages and in many cultures.

Set II. See the ancient illustration used in Was Pythagoras Chinese? (33-40). Also see Garfield’s Proof (41-47).

▶ The Theory of Area. The Area Postulate, Postulate 8, is a weak form of facts about area that are proven in calculus and analysis courses. The approach in such courses is to look at the Euclidean plane as the coordinate plane with the distance between points as the square root of the sum of the squares of the differences of the coordinates. This distance is based on the Pythagorean Theorem, which is built into the assumptions of the theory of area outlined below.

The inspiration for this theory is simple and practical: get upper and lower estimates of the area by laying a square grid over the region. This is the most natural approach to estimating the area of an irregular region.

The unit interval gives us the linear unit and it also gives us the unit square, the basic unit of area. The area of a region is estimated by placing a grid over the whole plane and finding the area of the squares of the grid falling wholly within the region (a lower bound on the region’s area) and the area of the squares of the grid needed to cover the entire region (an upper bound on the same area). If making the mesh finer and finer brings these upper and lower bounds as close as we please, then these bounds will converge to a common limit as the mesh size goes to zero. If so, then this common limit defines the region’s area. This notion of

area is called Jordan Content. See, for instance, Mathematical Analysis by Tom M. Apostol, Addison-Wesley, Reading, Massachusetts, 1957.

It is not possible to assign an area to every conceivable region using this definition. The reason is that making the mesh finer and finer will not necessarily bring these upper and lower bounds for the area arbitrarily close together. Increasingly zigzagged boundaries of fractal regions can keep the upper and lower bounds apart. But all polygonal regions will have areas, as would all regions of finite extent bounded by smooth curves. These areas will satisfy our postulates.

We can extend these ideas by covering our region by a collection of squares of any sizes we wish to use, perhaps using an infinite number of such squares, and then adding up the areas of these squares. There is no need to avoid overlapping of the squares, if we simply want an upper bound for the area of the region. The set of upper bounds for the region’s area obtained this way will be an interval of positive numbers, stretching to infinity, because we can increase the sizes of some of the squares in the covering and still get a covering. The lower end point of this interval is our best estimate of the area of the region, aR. If our region R sits within some rectangle T, consistency would require that aR + a(The points in T outside of R) = aT, where a gives the estimate of the area of any set gotten by looking at upper bounds. If this consistency principle holds, R is said to be measurable, and its area is aR. This notion of area satisfies our consistency conditions and it can be used to figure the areas of irregular plane sets such as fractals. The underlying ideas are simple, but technical difficulties make study of this notion of area, Lebesgue measure, a subject usually only treated in advanced courses. See, for instance, A Primer of Lebesgue Integration, H. S. Bear, Academic Press, San Diego, California, 1995. a