Mathematics
Finding the Area
Finding the area involves calculating the amount of space enclosed by a two-dimensional shape. The area of a shape is typically measured in square units, such as square meters or square inches. The process for finding the area varies depending on the shape, but generally involves multiplying the dimensions of the shape.
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eBook - PDF- Alberto D. Yazon(Author)
- 2019(Publication Date)
- Society Publishing(Publisher)
This chapter further explains the basic differences between area and perimeter. 4.2. DEFINITION OF AREA In the mensuration sector of mathematics, the area of a flat surface is being defined as the amount of space being covered by it. It is a physical quantity that indicates the number of square units being occupied by the two-dimensional object or a two-dimensional shape. It is being employed to know how much space is taken up by a flat surface. It is being measured in square units, such as square meters, square miles, square inches, etc. (Figure 4.2). Figure 4.2: Definition of area. Source: https://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Area. svg/1024px-Area.svg.png. The term area has various practical usages such as construction schemes, agriculture, architecture, and so on. To measure the area of a flat surface, one needs to count the number of squares being covered by the shape. Introduction to Mathematical Literacy 76 The area is the amount of two-dimensional space being taken up by the object. It is usually being measured in square units. As an example: Suppose if any individual person required tiling the floor of the room, the number of tiles needed to cover the complete room will be its area. 4.3. DEFINITION OF PERIMETER The perimeter is being explained as a measure of the length of the outline that surrounds a closed geometrical figure. The general term ‘perimeter’ is basically coming from the Greek word, ‘Peri’ and ‘meter’ which roughly translates to ‘around’ and ‘measure.’ In the field of geometry, it basically implies the continuous line creating the path outside the two-dimensional shape. The perimeter is the length all around the outside of a polygon or the course that encircles an area. This is different from the surface area. The surface area is how much surface is being encompassed by the polygon or space (Figure 4.3). Figure 4.3: Definition of the perimeter. Source: https://upload.wikimedia.org/wikipedia/commons/9/9b/Wet_perim-eter.PNG.
eBook - PDFGeometry Transformed
Euclidean Plane Geometry Based on Rigid Motions
- James R. King(Author)
- 2021(Publication Date)
- American Mathematical Society(Publisher)
For computing the area of other shapes, there are two paths one can take. The most general path is to define area by filling up the shape as much as possible with small squares and adding up the areas of the small squares. Taking ever smaller squares gives a larger and more accurate approximation of the area, much as we did for squares with irrational sides above. Taking a limit will produce a value for the area, at least for familiar geometric shapes. The full development of this general approach requires technical arguments that would be distracting from the main thread of this book, so we will only touch on this approach a few times, assuming some statements without proof. Instead, we will take an older and geometrically more interesting approach, which is to cut shapes apart and re-assemble them into other shapes whose areas are known. Area of Rectangles. Shapes that can be assembled from squares, like the rectan-gle on the left in Figure 2, will have an area equal to the sum of the area of the squares, in this case 3×4 = 12 . If squares are subdivided into smaller squares, such as a division into 4 smaller congruent squares, then the small squares have an area of 1/4 for each small square. Since the shaded figure inside the rectangle on the right contains 24 such squares, it has an area of 6. If a rectangle has sides ? and ? that are rational, then by choosing common denominators, one can divide such rectangles into squares and get the formula that area equals ? × ? . Areas of Triangles and Parallelograms 163 Figure 2. Area of Rectangle and of Figure Built from Squares For the general case, a limit argument is not hard, but there are direct proofs using dissection and the area formula for a square. In Figure 3, a rectangle ? with sides ? and ? is placed in a rotationally symmetric way into a square with sides ? + ? .
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
437 Geometric Figures: Areas and Volumes 6 OBJECTIVES After studying this unit you should be able to ● ● Express given customary area measures in larger and smaller units. ● ● Express given metric area measures in larger and smaller units. ● ● Convert between customary area measures and metric area measures. ● ● Compute areas, lengths, and widths of rectangles. ● ● Compute areas, bases, and heights of parallelograms. ● ● Compute areas, both bases, and heights of trapezoids. ● ● Compute areas of more complex figures (composite figures) that consist of two or more common polygons. As previously stated, in machine technology linear or length measure is used more often than area and volume measure. However, the ability to compute areas and volumes is required in determining job-material quantities and costs. Often, before a product is manufactured, part weights are computed. Volumes of simple geometric figures and combinations of figures (composite figures) must be calculated before weights can be determined. Section 6 presents area and volume measure of two-dimensional and three-dimensional geometric figures and practical area and volume applications. Areas of Rectangles, Parallelograms, and Trapezoids UNIT 59 CUSTOMARY AND METRIC UNITS OF SURFACE MEASURE (AREA) A surface is measured by determining the number of surface units contained in it. A surface is two dimensional. It has a length and width, but no thickness. Both length and width must be expressed in the same unit of measure. Area is computed as the SECTION SIX 438 SECTION 6 GEOMETRIC FIGURES: AREAS AND VOLUMES product of two linear measures and is expressed in square units. For example, 2 inches 3 4 inches 5 8 square inches (8 sq in. or 8 in. 2 ). The surface enclosed by a square that is 1 inch on a side is 1 square inch (1 sq in. or 1 in. 2 ). The surface enclosed by a square that is 1 foot on a side is 1 square foot (1 sq ft or 1 ft 2 ).
eBook - PDFIntroductory Mathematics
Concepts with Applications
- Charles P. McKeague(Author)
- 2013(Publication Date)
- XYZ Textbooks(Publisher)
• Set short-term and long-term goals. • Take breaks during your study sessions. • Begin each study session with the most difficult topics, then work through the easier topics toward the end of your session. • Revisit your study calendar to see if you have overbooked your time. • Change your time and location for studying. • Examine your diet and introduce healthier and more balanced meals. • Get adequate restful sleep between studying and going to class. • Schedule regular physical activity during your day. • Make use of your on-campus health services for either physical or psychological needs. OBJECTIVES 398 KEY WORDS Chapter 8 Geometry base width length height A Calculate the area of common geometric figures. 8.2 Area Imagine you are an artist who has just been asked to showcase your work in a local gallery. You have framed a number of pieces of varying sizes to display. To ensure they will all fit, you need to know the square footage of the wall where your artwork will hang. If you know that the wall is 30 feet long by 10 feet tall, how will you calculate its square footage? Square footage (ft 2 ) is one way to measure area. Recall that the area of a flat object is a measure of the amount of surface the object has. The area of the rectangle below is 8 square centimeters, because it takes 8 square centimeters to cover it. A Area As we have noted previously, the area of this rectangle can also be found by multiplying the length and the width. Area = (length) ⋅ (width) = (4 centimeters) ⋅ (2 centimeters) = (4 ⋅ 2) ⋅ (centimeters ⋅ centimeters) = 8 square centimeters From this example, and others, we conclude that the area of any rectangle is the product of the length and width. Here are the most common geometric figures along with the formula for the area of each one. The only formulas that are new to us are the ones that accompany the parallelogram and the circle.
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
D A B C 7 E 6 4 6 Figure 13.40 The area of the parallelogram is 6 4 24 ¥ = and the area of the triangle is 1 2 7 4 14 ¥ = for a total area of 38 square units. If the equation for the area of a trapezoid is used, the computation would be 1 2 6 13 4 2 19 38 ( ) + = = ¥ ¥ square units. ■ O A s B r Figure 13.41 Section 13.2 Length and Area 673 A rigorous verification of this formula cannot be done without calculus-level mathematics. However, areas of irregular two-dimensional regions can be approxi- mated by covering the region with a grid. Area of a Circle The area A of a circle with radius r is A r = p 2 . r Common Core – Grade 7 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. T H E O R E M 1 3 . 5 The dimensions of the “key” on a standard basketball court are shown in Figure 13.42 where the curve at the right is a semicircle. If you were given the task to paint the key prior to the lines being painted, how many square feet of area do you need to cover. 19 ft 12 ft Figure 13.42 SOLUTION This basketball key can be broken into a rectangle and a semicircle as shown in Figure 13.43. 19 ft 12 ft 6 ft Figure 13.43 The area of the rectangle is 12 19 228 2 ¥ = ft and the area of the semicircle is 1 2 2 6 18 56 5 2 p p ¥ ¥ = ≈ . ft for a total area of 284 5 2 . ft . ■ 674 Chapter 13 Measurement The Pythagorean Theorem The Pythagorean theorem, perhaps the most spectacular result in geometry, relates the lengths of the sides in a right triangle; the longest side is called the hypotenuse and the other two sides are called legs. Figure 13.44 shows a special instance of the Pythagorean theorem in an arrangement involving isosceles right triangles.
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
The purpose of this investigation is in the realm of “number sense,” helping you to see why A < 3 . 14 r 2 (and therefore A 5 p r 2 ) makes sense. Consider the circle in Figure 9.16(a), whose radius is r inches. In Figure 9.16(b), we have circumscribed a square around the circle. What is the area of the square? Determine this and then read on. . . . r r (a) (b) (c) Figure 9.16 DISCUSSION If the radius of the circle is r inches, then the length of each side of the square is 2 r , and thus the area of the square is H20849 2 r H20850H20849 2 r H20850 5 4 r 2 . Thus, the area of the circle is less than 4 times r 2 . We can use our spatial sense to estimate that the circle covers about 3 4 as much space as the square and thus approximate the area of the circle as 3 r 2 , or we can place a grid over the figure and determine what fraction of the square is covered by the circle. In this case, we get a more accurate estimate of the area of the circle (see Figure 9.16[c]). We can also derive the formula for the area of a circle by turning it into a shape that closely resembles a parallelogram. If we take a circle, cut it into sectors, and rearrange them, we get a shape that looks similar to a parallelogram. In the following figure, we cut the circle into 16 sectors. If we cut the circle into more sectors, we would get a shape that looks even more like a parallelogram. We already know that the area of a parallelogram is A 5 base 3 height. What are the base and the height in relation to the original circle? Think about this before reading on. . . . INVESTIGATION 9.2d Understanding the Area Formula for Circles Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.
eBook - PDFMeasurements from Maps
Principles and Methods of Cartometry
- D H Maling(Author)
- 2016(Publication Date)
- Butterworth-Heinemann(Publisher)
Using modern methods of data collection and storage there is no reason why they should not be used to measure the area enclosed by a coastline or other irregular boundary, even if this comprises many hundreds of thousands of separate coordinated points. 328 Measurements from Maps THE AREA FORMULAE OF PLANE GEOMETRY The formulae which follow are presented without proof, for all of them are well known and they are listed here to remind the reader how each of the measurements and calculations are made. The Area of a Triangle Figure 16.1(a) illustrates a triangle ABC in which one side is designated the base, b, and the line perpendicular to it which passes through the opposite vertex is called the height, h. Then, the area, A=b-h (16.1) If any of the angles of the triangle are also known, alternative solutions are A=absinC (16.2) (e) ( f ) FIG. 16.1. Measurement of area of simple plane figures according to the rules of Euclidean geometry: (a) the triangle ABC; (b) the rectangle ABCD; (c) the parallelo-gram ABCD; (d) the rhombus ABCD; (e) the trapezoid, ABCD; (f) any quadrilateral ABCD. Geometrical Methods of Area Measurement 329 (16.3) A =jbc'unA (16.4) For a triangle in which the perpendicular height had not been determined, and the angles have not been measured, but the lengths of all three sides are known, the half-perimeter formula may be used A = y J { s (s-a){s-b)(s-c)} (16.5) where s = ^ a + b + c) (16.6) or is one-half of the length of the perimeter of the figure. The Area of a Rectangle In Fig. 16.1(b) the rectangle ABCD has the sides AB = a and BC = b. Quite simply. A = a-b (16.7) The Area of a Parallelogram In Fig. 16.1(c) the figure has side AB = a and BC = b. Also h is the length of the perpendicular to one pair of the parallel sides. Then A = ah (16.8) Also A = abs'mB (16.9) The Area of a Rhombus The length of each side of the figure is equal to a.
eBook - PDF- Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
52. The perimeter of a rectangle is 32 cm. Where x is the width and y is the length, express the area A of the rectangle in terms only of x. 53. Square DEFG is inscribed in right uni25B3ABC, as shown. If AD 5 6 and EB 5 8, find the area of square DEFG. 54. TV bisects uni2220STR of uni25B3STR. ST 5 6 and TR 5 9. If the area of uni25B3STR is 25 m 2 , find the area of uni25B3SVT . 55. a) Find a lower estimate of the area of the figure by counting whole squares within the figure. b) Find an upper estimate of the area of the figure by counting whole and partial squares within the figure. c) Use the average of the results in parts (a) and (b) to provide a better estimate of the area of the figure. d) Does intuition suggest that the area estimate of part (c) is the exact answer? 56. a) Find a lower estimate of the area of the figure by counting whole squares within the figure. b) Find an upper estimate of the area of the figure by counting whole and partial squares within the figure. c) Use the average of the results in parts (a) and (b) to provide a better estimate of the area of the figure. d) Does intuition suggest that the area estimate of part (c) is the exact answer? Q P N M R S * * * * R S R S T A B D E G F C R T V S 362 CHAPTER 8 ■ AREAS OF POLYGONS AND CIRCLES 8.2 Perimeter and Area of Polygons We begin this section with a reminder of the meaning of perimeter. The perimeter of a polygon is the sum of the lengths of all sides of the polygon. DEFINITION Table 8.1 summarizes perimeter formulas for selected types of triangles, and Table 8.2 summarizes formulas for the perimeters of selected types of quadrilaterals. However, it is more important to understand the concept of perimeter than to memorize formulas. Study each figure so that you can explain its corresponding formula.
eBook - PDFDr. Math Introduces Geometry
Learning Geometry is Easy! Just ask Dr. Math!
- (Author)
- 2004(Publication Date)
- Jossey-Bass(Publisher)
It can really help you get a feel for how multiplication and factoring work! —Dr. Math, The Math Forum Dear Leon, You use a different formula because it is a different problem. The answer is different, so you need a different process to get the answer. (And you may find that they are not so different after all!) There is another way to explain this with a diagram. It helps to think of squares and rectangles, because if you think of right triangles in pairs, you can rearrange them to make squares or rectangles. And you already know how to find the area of squares and rectangles. Dear Dr. Math, Why do you have to use a different formula to get the area of a triangle than a rectangle or a square? Yours truly, Leon Areas of Triangles versus Areas of Rectangles 2 2 Because two identical right triangles put together along their long sides make a rectangle or a square, that means half the area of the square or rectangle is the area of the triangle. We write this as where b is the base and h is the height. When you are Finding the Areas of triangles that are not right tri- angles, try thinking about creating right triangles and applying this same idea. For example: This triangle becomes . . . two right triangles —Dr. Math, The Math Forum Units of Area When calculating area and perimeter, you’re dealing in units of measurement. It’s important to keep the units straight in your head while you work with them so that you don’t end up with the wrong units at the end of the process. Sometimes they’re hard to tell apart. Does 3 square meters mean (3 2 )m or 3 m 2 ? Which area is bigger: pi meters squared or 10,000 pi centimeters squared? This section will help you keep the units straight. Areas and Perimeters of Two-Dimensional (2-D) Geometric Figures 67 A b h = ⋅ 1 2
- Charlene E. Beckmann, Denisse R. Thompson, Rheta N. Rubenstein(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
How can you help students distinguish between these two values? e. The student page guides students through two methods to justify the formula for the area of a trapezoid. Discuss the pros and cons of providing students with multiple justifications of a result. f. Find another way to derive the area formula for a trapezoid. g. In justifying the formula for the area of a circle, some teachers will have students draw a circle, highlight the circumference, and then cut the circle into sectors and physically form the ‘‘bumpy parallelogram.’’ Why might such an approach be helpful when justifying this formula? 10. Choose one or more of the problems from Area Formulas for Familiar Two- Dimensional Figures to try with high school students. a. Review your responses to the problems on the student page (see DMP problem 9). For each problem, anticipate students’ work at three levels: misconceived, partial, and satisfactory. Record these responses in a QRS Guide. b. What teacher support in the form of questions will you be ready to provide for each recorded solution? Record appropriate teacher support questions in the QRS Guide. c. Try the problem(s) with high school students. How did they respond? Add any new responses to your QRS Guide. d. Where did students have difficulties? Include teacher support to help students overcome these difficulties. What reactions or difficulties surprised you? 2.5 Tasks with High Cognitive Demand: Measurement in the Plane and in Space 139 TEACHING AND LEARNING HIGH SCHOOL MATHEMATICS # 2010 John Wiley & Sons, Inc. Area Formulas for Familiar Two-Dimensional Figures 1. Parallelograms. On the geoboard template below, draw a nonrectangular parallelogram with a base of 4 units and a height of 5 units. Recall that the base is the length of one of the sides and the height is the length of the perpendicular segment drawn from the opposite side to the base. Label your parallelogram QUAD. a. Find the area of QUAD in square units.
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