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.
7 Key excerpts on "Finding the Area"
eBook - ePub- Surinder Virdi, Roy Baker, Narinder Kaur Virdi(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...CHAPTER 11 Areas (1) Learning outcomes: (a) Calculate the areas of triangles, quadrilaterals and circles (b) Identify and use the correct units (c) Solve practical problems involving area calculation 11.11 Introduction Area is defined as the amount of space taken up by a two-dimensional figure. The geometrical properties of triangles, quadrilaterals and circles have been explained in Chapter 10. A summary of the formulae used in calculating the areas and other properties of these geometrical shapes is given in Table 11.1. The units of area used in metric systems are: mm 2, cm 2, m 2 and km 2. Table 11.1 Shape Area and other properties Area = l × b Perimeter = 2 l + 2 b = 2(l + b) Area = l × l = l 2 Perimeter = 4 l Area = l × h Area = π r 2 Circumference = 2π r 11.2 Area of triangles There are many techniques and formulae that can be used to calculate the area of triangles. In this section we consider the triangles with known measurements of the base and the perpendicular height, or where the height can be calculated easily. Example 11.1 Find the area of the triangles shown in Figure 11.1. Figure 11.1 Solution: (a) Base BC = 8 cm We need to calculate height AD, which has not been given. As sides AB and AC are equal, BD must be equal to DC. Therefore, BD = DC = 4 cm. Now we can use Pythagoras’ Theorem to calculate height AD : Therefore (b) 11.3 Area of quadrilaterals A plane figure bounded by four straight lines is called a quadrilateral. The calculation of area of some of the quadrilaterals is explained in this section. Example 11.2 Find the area of the shapes shown in Figure 11.2. Figure 11.2 Solution: (a) Area of a rectangle = length × width Length = 15 cm, and width = 6 cm Area of rectangle ABCD = 15 × 6 = 90 cm 2 (b) In a square, the length is equal to the width...
eBook - ePubMathematics in the Primary School
A Sense of Progression
- Sandy Pepperell, Christine Hopkins, Sue Gifford, Peter Tallant(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...Whilst accepting that it should be taught meaningfully, it is not always necessary to take a lot of time and trouble concocting practical situations which pretend to contain area. Materials such as transparent grids and geoboards are useful for posing problems involving area. On geoboards ask children to make shapes with a particular area, for example, eight squares. Ask, ‘Look at the shapes. Will the perimeter remain the same?’ (Geoboards have nails at the points of a square grid, elastic bands are stretched between the nails to make shapes.) A common misconception is to overgeneralise the correct result that the area of a rectangle is obtained by multiplying the length by the breadth by applying this to all areas regardless of the shape in question. This can be avoided by ensuring that children find the area of many different kinds of shape, including irregular shapes such as leaves, and also insisting that children are precise in their use of mathematical statements, for example, ‘The area of a rectangle is length × breadth’. Perimeter Perimeter, an aspect of measuring length, is often confused with area. Dickson et al. (1984) suggest that this might be due to early formalisation through the introduction of formulae before children have had sufficient experience of exploring the shapes practically. They suggest activities to show that area can be varied while perimeter stays constant, and vice versa. If a shape has all its sides doubled but retains the same angles, its area will be quadrupled. Using squared paper or a geoboard the teacher can ask: ‘How many shapes can you make from 12 squares?’ ‘ What are the perimeters of those shapes? ’ ‘How many shapes can you make with a perimeter of 12?’ ‘Find the area of each of the shapes. Is it always the same?’ The perimeter of a circle is called the circumference. When using trundle wheels, children can be asked to measure the diameter of the wheel and compare this with the circumference...
eBook - ePub- Tandi Clausen-May(Author)
- 2013(Publication Date)
- SAGE Publications Ltd(Publisher)
...Learners spend time drawing shapes on squared paper, and counting and recording the number of squares used (the area), and the number of units around the edge (the perimeter). But this approach focuses on the numbers – and to a visual and kinaesthetic thinker one number may be very like another, so area and perimeter are likely to get muddled. But area and perimeter are quite different concepts. Perimeter is fairly straightforward. It is the distance around a shape. I can walk around the perimeter of a large shape, or trace my pencil around the perimeter of a smaller one – so I can see and feel what a perimeter is. But area is more difficult to understand. It may be thought of as ‘an amount of flatness’. Theme: Mathematical Language – Area and Perimeter The ‘mathematical’ terms area and perimeter may become easier to remember if they are associated with appropriate movements. Area may be thought of as a ‘measure of flatness’. A common sign for area is a hand held flat above the table, and moved round in a horizontal plane as if to smooth the air underneath. A perimeter is the distance around a shape. The common sign for this uses both hands. The forefinger of the left hand is held up, and then a roughly square path is sketched out in the air with the forefinger of the right hand. In the Classroom – Tiles and Sticks Activities that relate area and perimeter to different materials may provide a firmer foundation than mere counting for the development of these concepts. Square tiles, which can be picked up and moved around, provide a better starting point for area than drawn squares. A set of sticks that are the same length as the edge of a tile provide a model of the perimeter. The challenge may then be set to surround a given number of tiles with different numbers of sticks, or to fill different spaces, each surrounded by a given number of sticks, with different numbers of tiles...
eBook - ePub- Christine Hopkins, Ann Pope, Sandy Pepperell(Authors)
- 2013(Publication Date)
- David Fulton Publishers(Publisher)
...In measuring area (surface) the approach is to visualise the surface as a grid of squares which can be counted. In fact any tessellating shape will do because these cover the surface without gaps, ensuring all the surface is accounted for. However, squares are generally considered more convenient because they produce clear rows and columns, which means any shape can be thought of as a sum of rectangles, and the area of a rectangle can be calculated using multiplication. SCALES Nearly all aspects of measure involve the reading of some kind of scale, such as rulers, graded containers or kitchen scales. The continuous nature of measure is explicit on an analogue scale like a ruler and when reading scales the subdivisions of units are seen to be important in determining levels of approximation and accuracy. They are read ‘to the nearest…’. More sensitive scales which can represent very small units such as milligrams might be needed in some circumstances such as weighing out medicines while in others, such as buying food, weighing to the nearest 25 grams might be sufficient. With digital displays, however, the need to interpret scales is removed and the continuous nature of measure is less explicit because a discrete value is displayed. If a shopper asked for 500 g of fish at the supermarket the amount weighed will never be exactly that. It might show as 478 g on a digital display and cause the pointer on an analogue scale to move close to the 500 g mark and the shopper will need to decide whether they want fish a little over 500 g or a little under for their purposes. Mathematicians have devised methods of measuring the length, area and volume of increasingly complicated shapes. MEASURING DISTANCE PERIMETER The perimeter of a closed shape is the total distance round the edge of the shape. Perimeter of a circle The distance around a circle is known as the circumference. 1. Take a 1 × 1 square. 2. Fit a circle inside. 3. The length of the diameter is 1. 4...
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Chapter 7 The Shape of Things This chapter is about geometry. Every day, you see many things that have to do with geometry and you use geometric principles, even though you don’t think of them as geometry. Tires are circles, and they had better be attached at the exact center of the circle to function properly. Honeycombs are made up of hexagons (six-sided figures). Even the truss on a bridge is a trapezoid, and bridges are made up of many triangles because the triangles create rigidity. A lot of understanding geometry is knowing the words that describe a shape. Pay particular attention to the definitions in the following sections, although they are words you probably already know. Two words that pertain to all two-dimensional closed geometric figures are perimeter and area. (Closed means all the corners are connected.) The perimeter is the distance around a figure, or the sum of the lengths of all of its sides. A typical perimeter is a fence around a plot of land. Area is a term used for the space enclosed by any closed figure. It is expressed in square units (in 2, ft 2, and so forth) and is found by various formulas, some of which are on the GED ® test formula sheet. Typical areas that we see every day are a rug or a plot of land enclosed by a fence. Lines and Angles Geometric shapes have everything to do with lines and angles, so you must understand them first. Even circles, which themselves have no straight lines or angles, have straight lines and angles within them that tell, for example, the size of the circle as well as parts of the circle. A line actually goes on forever in both directions, or we say, “It goes on to infinity (∞) in both directions.” If we want to concentrate on a part of a line, we call that a line segment, and we show which line segment we mean by stating its endpoints. So if we are interested in a line that goes from the 1-inch to the 5-inch measure, we mean a 4-inch line segment...
eBook - ePubKnowing and Teaching Elementary Mathematics
Teachers' Understanding of Fundamental Mathematics in China and the United States
- Liping Ma(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...4 Exploring New Knowledge: The Relationship Between Perimeter And Area Scenario Imagine that one of your students comes to class very excited. She tells you that she has figured out a theory that you never told the class. She explains that she has discovered that as the perimeter of a closed figure 1 increases, the area also increases. She shows you this picture to prove what she is doing: How would you respond to this student? Students bring up novel ideas and claims in their mathematics classes. Sometimes teachers know whether a student’s claim is valid, but sometimes they do not. The perimeter and area of a figure are two different measures. The perimeter is a measure of the length of the boundary of a figure (in the case of a rectangle, the sum of the lengths of the sides of the figure), while the area is a measure of the size of the figure. Because the calculations of both measures are related to the sides of a figure, the student claimed that they were correlated. The immediate reactions of the U.S. and Chinese teachers to this claim were similar. For most of the teachers in this study, the student’s claim was a “new theory” that they were hearing for the first time. Similar proportions of U.S. and Chinese teachers accepted the theory immediately. All the teachers knew what the two measures meant and most teachers knew how to calculate them. From this beginning, however, the teachers’ paths diverged. They explored different strategies, reached different results, and responded to the student differently. How the U.S. Teachers Explored the New Idea Teachers’ Reactions to the Claim Strategy I: Consulting a Book While two of the U.S. teachers (9%) simply accepted the student’s theory without doubt, the remainder did not. Among the 21 teachers who suspected the theory was true, five said that they had to consult a book...
eBook - ePubProject Surveying
Completely revised 2nd edition - General adjustment and optimization techniques with applications to engineering surveying
- Peter Richardus(Author)
- 2017(Publication Date)
- Routledge(Publisher)
...Chapter 2 AREAS 2.1 Formulae In this chapter the formulae will be derived for the calculation of an area enclosed by a polygon the points of which have known coordinates. Suppose a triangle Δ ABC is given; the coordinates of the points A, B and C are X A, Y A ; X B, Y B ; and X C, Y C respectively (Fig. 2.1). Fig. 2.1 From analytic geometry the double area of a triangle ABC can be given in the form of a determinant 2 A 1 = | X A Y A 1 X B Y B 1 X C Y C 1 |, (2.1) where A 1 is the area. It is counted positive if the points A, B and C are taken clockwise. Expanding (2.1) gives 2 A 1 = X A (Y B − Y C) + X B (Y C − Y A) + X C (Y A − Y B), (2.2) which formula is cyclic. The area of a second triangle CBD (adjacent to Δ ABC) is in a similar. way 2 A 2 = X C (Y B − Y D) + X B (Y D − Y C) + X D (Y C − Y B), (2.3) so that the double area of the quadrilateral ABDC is given by the addition of (2.2) and. (2.3) 2 A = X A (Y B − Y C) + X B (Y D − Y A) + X D (Y C − Y B) + X C (X A -Y D). (2.4) By generalisation it can be proved, that the double area of any polygon of n points is given by the sum of the abscissae of a point multiplied by the difference of the ordinates of the following and the preceding point, counting clockwise. This theorem may be written in an abbreviated manner. as 2 A = [ X n (Y n+1 − Y n-1) ], (2.5) the square brackets indicating the summation. The formulae can also be given as the convenient expression 2 A = [ (X n Y n+1) − (Y n − 1 X n) ]. (2.6) By the expansion of the determinant according to the elements of the second row, it can be derived analogously that, counting anticlockwise + 2 A = [ Y n (X n+1 X n+1) ]. (2.7) The lay out of the formulae is shown in the Fig. 2.2 with the polygon ABCDE. As a check the sequence is again followed using the formula (2.7), starting with X E. If the number of points is even this sequence of manipulations cannot be carried out...
