Area of Parallelograms
The area of a parallelogram is the measure of the space enclosed within its boundaries. It is calculated by multiplying the base of the parallelogram by its height. The formula for finding the area of a parallelogram is A = base × height, where A represents the area.
6 Key excerpts on "Area of Parallelograms"
eBook - ePub- Tandi Clausen-May(Author)
- 2013(Publication Date)
- SAGE Publications Ltd(Publisher)
...Any other triangle can then be seen as a shearing of a right-angled triangle with the same base length, height and area. This ‘model to think with’ can be applied to any parallelogram or triangle, so that the learner can mentally transform the figure on the page back into a rectangle or right-angled triangle. Then the base and height can be identified easily, and so its area may be found. There is no need for a formula: the process is visual and kinaesthetic, not symbolic. PowerPoint PowerPoint 9-2, Areas of Straight-Sided Shapes, shows how to find the area of a parallelogram or a triangle by first shearing it so that it has a right angle, and then basing the calculations on the area of the resulting rectangle. e) Capacity and Volume We have seen that area can be thought of as ‘an amount of flatness’. It is a strictly two-dimensional concept. Capacity and volume, on the other hand, are three-dimensional. But our world is three-dimensional, so these concepts may actually be easier to understand. Capacity may be the best place to start. A capacity relates to a particular container, and it tells you how much that container can hold. This idea may be established using informal measures, such as the number of small cups-full that are needed to fill a big jug. Volume, on the other hand, is the amount of ‘stuff’ that is needed to make a solid. Activities which involve building up cuboids a layer at a time out of cubic-centimetre cubes to find their volumes are commonly used to introduce a more formal measure of volume. This is a good practical approach, and it may be extended in due course to other solids – but it is important to emphasise that the volume relates to the whole block of cubes, not to an empty container. An empty carton has a capacity – but it is empty, so its only ‘volume’ is the volume of plastic or cardboard of which it is composed...
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...
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...Since both opposite sides are congruent and parallel, we can further classify it by saying “parallelogram.” Circles Circles are not polygons. Although a circle is a closed-plane figure, it is not made from segments. A circle is the locus of all points in a plane having the same distance from a fixed point. The fixed point is called the center of the circle. The distance between the center of the circle and any point on the circle is called the radius. The distance from any point on the circle through the center to another point on the circle is called the diameter. The distance around the circle is called the circumference. EXAMPLE 11.4 Classify the polygons by the number of sides and, if they are triangles or quadrilaterals, use their sides and angles to give a more specific classification. 1) 2) 3) 4) 5) 6) 7) SOLUTIONS 1) hexagon 2) acute triangle 3) decagon 4) rhombus 5) trapezoid 6) scalene triangle 7) square 11.5 What Are Area and Perimeter and How Do We Calculate Them? The area of a shape is the number that tells you how many square units are needed to cover the shape. Area can be found in different units, such as square feet, square meters, or square inches. EXAMPLE: This can be illustrated by using a square grid of centimeters; however, there are formulas you can use to calculate area. You need to be careful to make sure you are using the correct dimensions. The height must be perpendicular to the base. You can find the area of many polygons by knowing these formulas and constructing or deconstructing the polygon into these shapes. EXAMPLES: Given a square with sides of 9 in: Find the area. A = s 2 A = (9) 2 A = 81 in 2 Given a parallelogram with a height of 9.6 feet and a base of 16.2 feet: Find the area. A = bh A = (9.6)(16.2) A = 155.52 feet 2 For example, the area of the trapezoid below can be found by using the trapezoid formula or by deconstructing the trapezoid into a rectangle and a triangle. The results will be the same...
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 - 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 - 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...
