The sound of a drum is based on the wave equation from physics. The surface area of the
drumhead and its fixed perimeter are key parameters. This paper uses area and perimeter to model and
answer the question, Can a rectangular and a circular drum make the same sound

Adabor, James K.

Foley, Gregory D.

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Shape Affects the Sound ofa Drum: Modeling Area andPerimeterJames K. Adabor, Wright State UniversityGregory D. Foley, Ohio UniversityAbstract: The sound of a drum is based on the wave equation from physics. The surface area of thedrumhead and its fixed perimeter are key parameters. This paper uses area and perimeter to model andanswer the question, Can a rectangular and a circular drum make the same sound?Keywords. measurement, calculus, physics, modeling, multiple representations1 IntroductionThe trained ear can hear the shape of a drum! The sound of a drum is based on the area andperimeter of the drumhead, and the tension of the drumhead’s skin. Mark Kac (1966) openeda debate regarding whether it is always possible to hear the shape of a drum. In 1991, CarolynGordon and her colleagues “found two drums that have different shapes yet exactly the same soundwhen played” (Greenwald & Nestler, 2001, p. 12), but these were no ordinary drums. For practicalpurposes, the shape of a drumhead determines the sound of the drum. And in all cases, the sounddepends on the area and perimeter of the drumhead. Greenwald and Nestler got us interested inthe shapes and sounds of drums and how this topic might be an enticing way to get students tothink deeply about the ideas of area and perimeter.Understanding area and perimeter - and knowing which is which - is important for students inGrades 3-12 and beyond. For example,• In Grade 3, students are expected to “recognize perimeter as an attribute of plane figures anddistinguish between linear and area measures” (National Governors Association Center forBest Practices & Council of Chief State School Officers, 2010, p. 22).• In Grade 4, they “apply the area and perimeter formulas for rectangles in real world andmathematical problems” (p. 31).• In Grade 7, students should “know the formulas for the area and circumference of a circle anduse them to solve problems” (p. 50).Greenwald and Nestler (2001) made a statement that especially intrigued us: They claimed “that acircle and a rectangle can never have both the same area and the same perimeter” (p. 14). Greenwaldand Nestler suggested verifying this claim using computer algebra and plotting functions in threedimensions, but we have found a high school-friendly approach that we wish to share with teachersand teacher educators. We think that high school students could investigate Greenwald andOhio Journal of School Mathematics 68 Page 1Nestler’s claim in a variety of ways and with a variety of tools. Moreover, such an investigationwould reinforce and deepen their understanding of the area and perimeter of rectangles and circleswhile developing their reasoning and problem-solving skills.The purpose of this article is to show some possible solution paths. In doing so, we hope tostimulate the creation of instructional tasks that integrate the high school conceptual categoriesof algebra, functions, geometry, and modeling with mathematical processes (NCTM, 2000) andmathematical practices (NGA Center & CCSSO, 2010). Our investigation of this claim about a drumbegins with quantitative reasoning in the geometric realm, but quickly moves to abstract reasoningin the world of algebra and functions, and finally returns to the original context of the shape andsound of a drum. Thus, it is a nice example of the Common Core’s modeling cycle, shown in Fig 1.Fig. 1: A schematic diagram of mathematical modeling (NGA Center & CCSS), 2010, p. 72).2 A Rectangle and a Circle Can’t Have Both Equal Areas and EqualPerimeters2.1 Formulating the ProblemWe assume equal perimeters and equal areas and seek a contradiction, noting that the perimeter ofa circle is its circumference. Let the side lengths of the rectangle be the positive numbers x and y,and let radius of the circle be the positive number r, all three measured using the same units. Thenthe perimeter of the rectangle = 2(x+ y), and the circumference of the circle = 2pir. The assumptionthat the rectangle and the circle have equal perimeters implies that2pir = 2(x+ y)⇔ r = x+ ypi. (1)If, in addition, the rectangle and circle have equal areas, thenpir2 = xy ⇔ r =√xypi. (2)Equating the r values in equations (1) and (2) yieldsx+ ypi=√xypi⇔ x+ y = pi√xy√pi⇔ x+ y = √pixy. (3)The question becomes, Does equation (3) have any solutions for positive values of x and y? If so, wehave found another solution to the problem posed by Mark Kac in 1966. If not, we have found thecontradiction we seek.Page 2 Ohio Journal of School Mathematics 682.2 Reformulation and ComputationWe now try to solve equation (3). We use a graphical approach and begin by letting y = b for anarbitrary positive real number b. Thus, equation (3) becomes x+ b =√pixb. We then define twofunctions: f1(x) = x+ b and f2(x) =√pib√x. (By substituting b for y, these are now functions ofone variable instead of two, which makes the problem accessible to high school students.) Thegraphs of these functions are shown in Fig 2.Try creating these graphs yourself for various positive values of b. In all cases, it appearsthat f1(x) > f2(x) and that the essential relationship between the two graphs is unchanged as bvaries. The positive number b acts as a dilation factor with (0, 0) serving as the center of dilation.Said another way, any two versions of Fig 2 that are obtained by using different values of b aregeometrically similar to one another. (This is a nice connection between the world of functiongraphing and the world of plane geometry.) Although this visual evidence is convincing, it fallsshort of the contradiction needed for a formal proof.Fig. 2: The graphs for f1 and f2 with a scale of b units on both axes.2.3 A Second ReformulationWe can take the graphical approach one step further by graphing f3(x) = f1(x)− f2(x),as shownin Fig 3. As may be expected at this point, regardless of the particular value of b > 0, it appearsthat the graph of y = f3(x) is strictly positive. Furthermore, Fig 4 illustrates that, for the caseb = 1, the minimum feature of the graphing software indicates that the least value of f3 is about0.215, which occurs when x = 0.785. By dilation, the minimum point for the general case wouldbe approximately (0.785b, 0.215b). Once again, we have strong evidence of a contradiction of ouroriginal assumption that a circle and a rectangle exist that have equal areas and equal perimeters. But westill do not have a proof.2.4 Modeling With CalculusA proof that f3(x) = x+b−√pib√x is strictly positive throughout its domain requires some calculus.So, for completeness, we now confirm with certainty that the minimum value of f3 is positive. Webegin by setting the derivative of this function to zero:f ′3(x) =[1−√pib2√x= 0]Ohio Journal of School Mathematics 68 Page 3Fig. 3: The graph of f3 with a scale of b units on both axes.Fig. 4: The graph of f3 with its minimum point shown for the case b = 1.We then solve the bracketed equation:√pib2√x= 12√x =√pib4x = pibx =pib4Notice that the second derivative f ′′3 (x) =√pib4√x3is undefined at 0, where a local maximum occurs,and is strictly positive on the rest of the domain of f3, (0,∞), and hence the critical value of x = pib4will produce an absolute minimum value of the function:f3(pib4)=pib4+ b−√pib√pib4=pib4+ b− pib2= b− pib4= b(1− pi4)because 4 > pi, pi4 < 1, and 1 − pi4 > 0. Because b is also positive, the minimum value of f3 is apositive number and f3(x) is never zero. Thus x+ y =√pixy has no solution, and it is impossiblefor a circle and a rectangle to be both equal in area and in perimeter.Page 4 Ohio Journal of School Mathematics 682.5 Reporting the Results and Looking BackThus we conclude that a rectangle and a circle can never have both the same area and the sameperimeter. Returning to the original context, a rectangular drum will never sound like a rounddrum! This conclusion was reached using multiple representations; it was demonstrated by aprecalculus graphical approach and formally proved using calculus techniques. Before moving onwe note that• Reducing the number of independent variables from two to one was key; this avoided theneed for three-dimensional graphing and other techniques beyond the high school curriculum.• The basic fact that 4 > pi was crucial in the formal proof.• As Fig 5 shows, if we check the calculus solution against the precalculus solution, we obtainconsistent results.Fig. 5: A comparison of approximate and exact results.3 Final ThoughtsThe “shape of a drum” problem that we have solved should be presented to students in a waythat encourages reasoning and sense making by requiring the “students to figure things out forthemselves” (NCTM, 2009, p. 11). By putting the onus on the students in this manner, the drumproblem has the potential to address the Standards for Mathematical Practice if students• make sense of the problem,• persevere in solving it,• combine quantitative and abstract reasoning,• construct logical arguments and use precise thinking,• employ mathematical models and the modeling process, and• “use appropriate tools strategically” (NGA Center & CCSSO, 2010, pp. 78).Ohio Journal of School Mathematics 68 Page 5In addition, we hope that our paper helps you give students an opportunity to connect geometry,algebra, and functions, to deepen their proficiency in reasoning and proof, and to solidify importantideas about area and perimeter. We encourage you to think of ways to provide some scaffoldingfor students while they work on this problem without spoiling their potential for invention anddiscovery. Moreover, we look forward to hearing how you have used the drum problem in yourclassroom.ReferencesGreenwald, S. J., & Nestler, A. (2001). Hearing the shape of a drum. The Centroid, 28(1), 1115. Seealso http://mathsci2.appstate.edu/~sjg/ncctm/activities/#drumKac, M. (1966). Can one hear the shape of a drum? American Mathematical Monthly, 27(4), 123.doi:10.2307/2313748National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.Reston, VA: Author.National Council of Teachers of Mathematics. (2009). Focus in high school mathematics: Reasoning andsense making. Reston, VA: Author.National Governors Association Center for Best Practices & Council of Chief State School Officers(2010). Common core state standards for mathematics. Washington, DC: Author. Retrieved fromhttp://corestandards.org/assets/CCSSI_Math%20Standards.pdfJames K. Adabor, james.adabor@wright.edu, is an Assistant Profes-sor of Mathematics Education at Wright State University, Lake Campus.His interests include the use of technology in inquiry-based mathemat-ics instruction and learning thereby promoting pedagogical skills amongearly and middle childhood pre-service teachers.Gregory D. Foley, foleyg@ohio.edu, is the Robert L. Morton Profes-sor of Mathematics Education at Ohio University. He advocates forinquiry-based methods and innovative uses of technology for mathe-matics teaching and learning at all levels, especially in Grades 6-14.Page 6 Ohio Journal of School Mathematics 68

Shape Affects the Sound of a Drum: Modeling Area and Perimeter

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Shape Affects the Sound of a Drum: Modeling Area and Perimeter

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