Area of Rectangles
The area of a rectangle is the measure of the space inside the shape, calculated by multiplying the length and width of the rectangle. The formula for finding the area of a rectangle is A = l * w, where A represents the area, l is the length, and w is the width. This concept is fundamental in geometry and is used in various real-world applications.
7 Key excerpts on "Area of Rectangles"
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- 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- 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...
- 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 - ePubKnowing and Teaching Elementary Mathematics
Teachers' Understanding of Fundamental Mathematics in China and the United States
- Liping Ma(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...The problem of the student was that she only saw the first one. Theoretically, with the same perimeter, let’s say 20 cm, we can have infinite numbers of rectangles as long as the sum of their lengths and widths is 10 cm. For example, we can have 5 + 5 = 10, 3 + 7 = 10, 0.5 + 9.5 = 10, even 0.01 + 9.99 = 10, etc., etc. Each pair of addends can be the two sides of a rectangle. As we can imagine, the area of these rectangles will fall into a big range. The square with sides of 5 cm will have the biggest area, 25 square cm, while the one with a length of 9.99 cm and a width of 0.01 cm will have almost no area. Because in all the pairs of numbers with the same sum, the closer the two numbers are, the bigger the product they will produce … (Tr. Xie) Tr. Xie and Tr. Mao did not draw on the same basic principles of mathematics for their arguments. However, both developed solid arguments. In fact, a basic principle of mathematics may be able to support various numerical models. On the other hand, a numerical model may also be supported by various basic principles. A profound understanding of a mathematical topic, at last, will include certain basic principles of the discipline by which the topic is supported. Passing through various levels of understanding of the student’s claim, the teachers got closer and closer to a complete mathematical argument. A Map of How Teachers’ Exploration Was Supported The teachers explored the student’s claim and reached an understanding of the mathematical issues at various conceptual levels: finding a counter-example, identifying the possible relationships between area and perimeter, clarifying the conditions under which those relationships hold, and explaining the relationships. While in the three previous chapters we were interested in teachers’ existing knowledge of school mathematics, now we are interested in their capacity for exploring a new idea...
eBook - ePubBasic Math Concepts
For Water and Wastewater Plant Operators
- Joanne K. Price(Author)
- 2018(Publication Date)
- Routledge(Publisher)
...9 Linear Measurement SKILLS CHECK Complete and score the following skills test. Each section should be scored separately in the box provided to the right. A score of 4 or above indicates you are sufficiently strong in that concept. A score of 3 or below indicates a review of that section is advisable. 9.1 Perimeter Calculations SUMMARY 1. To determine the perimeter (the distance around) of any angular area or object, add the length of each side. This is the general perimeter equation. Perimeter = Length + Length + Length + Length + … Side 1 Side 2 Side 3 Side 4 P = l 1 + l 2 + l 3 + l 4 + … 2. To determine the perimeter of a square, you may use the general equation above, or the modified equation listed below: P = 4 s 3. To determine the perimeter of a rectangle, you may use the general equation or the modified equation as follows: P = 2 l + 2 w Linear measurement is simply the measurement along a line. These lengths may be expressed using the English System of measurement, such as inches, feet, yards, and miles, or using the Metric System of measurement, such as millimeters, centimeters, meters, and kilometers. ∗ Many water and wastewater calculations require tank or channel dimensions, pipe lengths and diameters, weir lengths, and other linear measurements. Although these dimensions are normally provided in the treatment system plans and specifications, it may be wise to verify the lengths indicated. This chapter focuses on one particular type of linear measurement: the distance around the outside edge of an area or object— the perimeter and circumference. ADD LENGTHS OF SIDES The distance around an angular object or area is called the perimeter. To calculate the perimeter of any area or object add the length of each of its sides: Perimeter = l 1 + l 2 + l 3 + … The number of terms added for the perimeter equation depends on how many sides the object has...
