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AREA and VOLUME WHERE DO the FORMULAS COME FROM? Roger Yarnell John Carroll University, [email protected]

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John Carroll University Carroll Collected Masters Essays Theses, Essays, and Senior Honors Projects Spring 2016 AREA AND VOLUME WHERE DO THE FORMULAS COME FROM? Roger Yarnell John Carroll University, [email protected] Follow this and additional works at: http://collected.jcu.edu/mastersessays Part of the Geometry and Topology Commons Recommended Citation Yarnell, Roger, "AREA AND VOLUME WHERE DO THE FORMULAS COME FROM?" (2016). Masters Essays. 33. http://collected.jcu.edu/mastersessays/33 This Essay is brought to you for free and open access by the Theses, Essays, and Senior Honors Projects at Carroll Collected. It has been accepted for inclusion in Masters Essays by an authorized administrator of Carroll Collected. For more information, please contact [email protected]. AREA AND VOLUME WHERE DO THE FORMULAS COME FROM? An Essay Submitted to the Office of Graduate Studies College of Arts & Sciences of John Carroll University In Partial Fulfillment of the Requirements For the Degree of Masters of Arts in Mathematics By Roger L Yarnell 2016 This essay of Roger L. Yarnell is hereby accepted: ________________________________________________ __________________ Advisor – Douglas Norris Date I certify that this is the original document ________________________________________________ __________________ Author – Roger L. Yarnell Date TABLE OF CONTENTS 1. Introduction 2 2. Basic Area Formulas 4 3. Pi and the Circle 10 4. Basic Volume Formulas 14 5. Sphere 17 6. Pythagorean Theorem 20 7. Conclusion 23 8. References 24 1 1. INTRODUCTION What are area and volume? This was a question that was posed to incoming geometry students that have previously worked with these measurements in middle school. There were a variety of responses to the question, how are area and volume defined? Approximately half of the students had a general idea of what area and volume measure. The students stated that area is the amount of space in a two dimensional figure and volume is the amount of space in a three dimensional object, the students made no mention as to the units used to measure area and volume. Only one student stated that area is measured in square units and volume in cubic units. Other student responses for area, included multiplying two sides (the length times the width) and the length around an object. Yet other student responses for volume, included the width and length combined, the mass of an object, the weight of an object, how much area is in an object, and it is found by multiplying length, width and height together. Why are there so many different misconceptions for area and volume? Part of the reason for this may be that the students are given the formulas and are asked to simply compute the area and volume of different shapes and objects by substituting in values and plugging it into a calculator to get an answer. This does not give any fundamental understanding as to what the students are calculating or give them the meaning as to how this formula works for the selected object. As a result, students do not fully understand what their answer represents nor does it have any real or true meaning. The students need to know the meaning of area and volume and where the concepts originate in order to give them meaning. When the students understand where concepts come from they can obtain a deeper understanding and this will allow them to know the meaning of their solution. The students have been shown why the area of a rectangle is length times width but not for other shapes. This is why the students remember and think area is simply length times width. The students use the other formulas not knowing why those formulas work to find area for a particular figure. As the instructor, I was not proving all the formulas for area and volume of objects. I would prove some of the formulas, such as area of a rectangle, square, parallelogram, 2 triangle, and trapezoid and the volume of a rectangle prism. I thought these could be done with complete understanding but not the other formulas with more complicated proofs. Due to time constraints, the decision was made that it was more important for the students to apply the formulas to problem solving. Problem solving is a very important skill, but after further education in the graduate courses, it was revealed how important it is for the students to understand where the formulas for area and volume originate and how exactly the formulas give the desired result. This is a unit that includes the history of area and volume and will demonstrate and prove the formulas of each figure and shape that are used to calculate both area and volume. This unit will have the student gain an understanding of the mathematics behind both area and volume. This unit will also further enhance an appreciation and memory of what the students are calculating and understand the meaningfulness of their answer. What are area and volume? They are defined as follows [6]: AREA is the number of unit square inside a closed region. VOLUME is the number of unit cubes in a solid figure. The earliest beginnings of geometry can be traced back to around 3000 BC to ancient Egypt and Mesopotamia. They had used lengths, angles, areas, and volumes for construction and surveying purposes. [5] The Babylonian and Egyptian people knew the basic formulas for area and volume. They had descriptions for the area of several figures, some correct and for others, such as the triangle, we are not sure they were completely correct. They regarded geometry as a practical tool for their civilization. It is believed that they did not have deductive proofs of their formulas. [7] 3 2. BASIC AREA FORMULAS The area of a rectangle is probably the easiest to visualize. When the rectangle is divided into unit squares the formula for area can be developed. By counting these unit squares, the area of the rectangle can be determined. Also, the value for area can be found by doing repeated addition. Count the number of squares in each row, the number of rows within the rectangle, and the product will be the area. As in the diagram, there are five squares in a row and this is to be counted two times because there are two rows. Repeated addition in this manner is multiplication, so the number of squares can be found by multiplying the dimensions of length and width as this method of counting is a definition of multiplication. Therefore, the area of a rectangle can be defined as length multiplied by width. AREA OF A RECTANGLE: A = LW The area of a square can also be found by applying the formula for the area of a rectangle. A square is a rectangle with four congruent sides. If each side of the square is represented by the variable S, using the formula A=lw and substituting S for both length and width, then A = S∙S = S2 is obtained. Therefore the formula for the area of a square is A = S2. AREA OF A SQUARE: A = S2 The formula for the area of a parallelogram can be derived from the area of a rectangle by taking the right triangle from one side of parallelogram and placing it onto the other side to form a rectangle. As shown in the diagram on the next page of the parallelogram ABED, the right triangle ABC can be removed and placed on the right side 4 forming right triangle DEF. The length BE is the base of the parallelogram with AC being its height. After moving triangle ABC to the position of DEF a rectangle of equal area is formed. Segment AC is equal to DF which are the height and also the width of the rectangle ADFC. The base BE is also congruent to CF since BC and EF are congruent. BE = BC + CE, by the segment addition postulate and BE = CE + EF by substitution. CF = CE + EF, so by substitution BE = CF. The base of the parallelogram is equal to the width of the rectangle. Therefore, since the area of a rectangle is A = LW and the base of the rectangle is equal to the length of the rectangle and the height of the parallelogram is equal to the width of the rectangle the area of a parallelogram is equal to base times height. A = bh. AREA OF A PARALLELOGRAM: A = bh The formula for the area of a triangle can be derived by drawing a diagonal in a parallelogram. The opposite sides of a parallelogram are congruent. Using D C parallelogram ABCD, in the diagram to the right, by drawing diagonal BD we create two congruent triangles ABD and CDB. The tringles are congruent using the SSS postulate, since the opposite sides of a A E B parallelogram are congruent and BD = BD by the reflexive property. Since the triangles are congruent the area of one triangle is equal to half of the area of the parallelogram. Therefore the area of a triangle is equal to half of the base times its height. A = ½bh. AREA OF A TRIANGLE: A = ½bh Below is an activity that the students can complete to prove the formulas for the areas of a parallelogram and triangle by using grid paper, a ruler, and scissors. 5 1. Draw two large parallelograms using the condition: If two line segments are parallel and congruent then the quadrilateral formed is a parallelogram. Make two of the sides horizontal. 2. Draw two altitudes in each parallelogram from a vertex to a horizontal side. 3. Label each horizontal side with a “b” and each height with an “h”.
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