Area and Perimeter of Quadrilaterals
The area of a quadrilateral is the measure of the space inside the shape, calculated using various formulas depending on the type of quadrilateral. The perimeter of a quadrilateral is the total length of its sides. For example, the area of a rectangle can be found using the formula length × width, while the perimeter is calculated by adding all four sides together.
6 Key excerpts on "Area and Perimeter of Quadrilaterals"
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- 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 - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...A quadrilateral is any polygon with four sides. The points where the sides meet are called vertices (sin gular: vertex). PARALLELOGRAMS A parallelogram is a quadrilateral whose opposite sides are parallel. Two angles that have their vertices at the endpoints of the same side of a parallelogram are called consecutive angles. So A is consecutive to B ; B is consecutive to C ; C is consecutive to D ; and D is consecutive to A. The perpendicular segment connecting any point of a line containing one side of a parallelogram to the line containing the opposite side of the parallelogram is called the altitude of the parallelogram. A diagonal of a polygon is a line segment joining any two nonconsecutive vertices. The area of a parallelogram is given by the formula A = bh, where b is the base and h is the height drawn perpendicular to that base. Note that the height is the same as the altitude of the parallelogram. Example: The area of the parallelogram below is: A = bh A = (10)(3) A = 30 RECTANGLES A rectangle is a parallelogram with right angles. • The diagonals of a rectangle are equal,. • If the diagonals of a parallelogram are equal, the parallelogram is a rectangle. • If a quadrilateral has four right angles, then it is a rectangle. • The area of a rectangle is given by the formula A = lw, where l is the length and w is the width. Example: The area of the rectangle below is: A = lw A = (4)(9) A = 36 RHOMBI A rhombus (plural: rhombi) is a parallelogram that has two adjacent sides that are equal. • All sides of a rhombus are equal. • The diagonals of a rhombus are perpendicular bisectors of each other. • The area of a rhombus can be found by the. formula where d 1 and d 2 are the diagonals. ABCD is a rhombus. AC = 4 and BD = 7...
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...
- Fuchang Liu(Author)
- 2017(Publication Date)
- Routledge(Publisher)
...In a rectangle where the two pairs of sides are different in their linear distances (I’ve intentionally avoided using length here), length is one dimension and width is the other. It really doesn’t matter one way or the other. It doesn’t make any difference either. For example, suppose the longer side of a rectangle is 8 centimeters and the shorter side is 5 centimeters. If we name the longer side “length” (l = 8) and the shorter side “width” (w = 5), then the perimeter of this rectangle is p = 2 l + 2 w = 2 × 8 + 2 × 5 = 16 + 10 = 26 cm, and its area is a = lw = 8 × 5 = 40 cm 2. Alternatively, if we name the shorter side length (l = 5) and the longer side width (w = 8), then the perimeter is p = 2 l + 2 w = 2 × 5 + 2 × 8 = 10 + 16 = 26 cm, and its area is a = lw = 5 × 8 = 40 cm 2. Not a thing has come up differently. Figure 10.1 A 5 × 5 Square The perimeter and area of a square can be calculated using the formulas for a rectangle because it is a rectangle. A look at this problem in a different perspective will tell us why it shouldn’t matter one way or the other. Suppose there were such a stipulation that the longer sides were called “lengths” and the shorter sides “widths.” How could we accommodate a square? We know a square is a special rectangle and—while it has its own perimeter and area formulas for easier calculation purposes (p = 4 s and a = s 2, where s refers to the side)—any formula for a rectangle should apply to a square as well. For example, if a square has a side of 5 cm (see Figure 10.1), then its perimeter and area are p = 4 s = 4 × 5 = 20 cm and a = s 2 = 5 2 = 25 cm 2. Then, by way of formulas for a rectangle, p = 2 l + 2 w = 2 × 5 + 2 × 5 = 10 + 10 = 20 cm and a = lw = 5 × 5 = 25 cm 2 (in any square, l = w = s), respectively. This should hardly come as a surprise, because if formulas for rectangles didn’t apply to squares, then squares would no longer be rectangles...
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...
