Area of Plane Figures
The area of plane figures refers to the measure of the space enclosed by a two-dimensional shape, such as a square, rectangle, triangle, or circle. It is calculated by using specific formulas for each shape, such as base times height for a triangle or side squared for a square. Understanding the concept of area is essential for solving geometric problems and real-world applications.
6 Key excerpts on "Area of Plane Figures"
- 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 - ePub- Tandi Clausen-May(Author)
- 2013(Publication Date)
- SAGE Publications Ltd(Publisher)
...Geometry and Measures are as badly affected in this respect as any other area of the curriculum. Test and examination papers commonly include a Formula Sheet, in which all the learner’s understanding of the concepts of area and volume are reduced to a set of rules. Even when this sheet is not provided learners may still be taught the formulas by rote, rather than developing an understanding of the mathematics that underlies them. Then, of course, they forget them. So here again, learners who think more easily in pictures than in words and symbols need ‘models to think with’. The models make sense, so they are memorable – unlike the formulas, which are eminently forgettable. And having remembered the relevant model, the ‘picture in the mind’, the learner can work out the formula they need for the problem they are working on. d) Areas of Straight-Sided Shapes The first formula learners are likely to meet is for the area of a rectangle. They start by finding the areas of small rectangles by counting squares – that is, by yet more sequential recitation of disconnected number words. This follows logically from the standard introduction to Number, relying heavily on counting, that was discussed in Chapter 2, but it is less helpful for learners who see the whole picture at once. On the other hand, visualising a rectangle, and finding its area, lies at the heart of the area model of multiplication discussed in Chapter 3. Learners who see the calculation 3 × 4, for example, as have no need of a formula. They already understand the relationship between the edge lengths and the area of a rectangle on which the formula is based. The area of a parallelogram can be worked out directly from the area of a rectangle...
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...
eBook - ePub- Mark Zegarelli(Author)
- 2011(Publication Date)
- For Dummies(Publisher)
...Part IV Visualizing Plane Geometry and Trigonometry In this part. . . I n the chapters in this part, I cover plane geometry and trigonometry. I help you sharpen your skills on lines and angles, triangles, quadrilaterals, circles, solids, trig ratios, matrices, logarithms, and imaginary numbers. I also include some practice problems to get you up to speed quickly. Chapter 10 Plain Talk about Plane Geometry In This Chapter Answering questions involving angles Familiarizing yourself with triangles Making peace with quadrilaterals Discovering the ins and outs of circles Looking into solid geometry I n this chapter, the topic is geometry. Although a typical geometry course spends a lot of time focusing on geometric proofs, what you need to succeed on the ACT is a review of the basics. The chapter begins with a refresher on angles, including vertical angles, supplementary angles, angles that involve parallel lines, and angles in a polygon. Then I move on to triangles. I discuss the area formula for a triangle, the Pythagorean theorem, and a variety of common right triangles that you’re sure to see on the ACT. Next, I move on to quadrilaterals, showing you the formulas for the area and perimeter of squares and rectangles as well as the area of a parallelogram and a trapezoid. I also show you how to find the area and perimeter of a circle and how to answer ACT questions that involve tangent lines, arc length, and chords. I round out the chapter with a discussion of solid geometry, which is the extension of plane geometry to three dimensional solids. I introduce you to the formulas for the volume and surface area of a cube, a box (rectangular solid), and a sphere. I also give you a few formulas for a variety of other common solids. Knowing Your Angles In your geometry class, you probably spent a lot of time proving theorems about angles. This is good training for clear thinking and, of course, for passing your geometry tests...
eBook - ePub- KJ Rawson, E. C. Tupper, E. C. Tupper(Authors)
- 2001(Publication Date)
- Butterworth-Heinemann(Publisher)
...The area of a surface in the plane of O xy defined in Cartesian co-ordinates, is A = ∫ y d x in which all strips of length y and width δx are summed over the total extent of x. Because y is rarely, with ship shapes, a precise mathematical function of x the integration must be carried out by an approximate method which will presently be deduced. Fig. 2.13 There are first moments of area about each axis. (For the figures shown in Fig. 2.14, x 1 and y 1 are lengths and x and y are co-ordinates.) M y y = ∫ x y 1 d x and M x x = ∫ x 1 y d y Fig. 2.14 Dividing each expression by the area gives the co-ordinates of the centre of area, (x ¯, y ¯) : x ¯ = 1 A ∫ x y 1 d x and y ¯ = 1 A ∫ x 1 y d y For the particular case of a figure bounded on one edge by the x -axis * M y * = ∫ 1 2 y 2 d x and y ¯ = 1 2 A ∫ y 1 2 d x For a plane figure placed symmetrically about the x -axis such as a. waterplane, M x x = ∫ x 1 y dy = 0 and the distance of the centre of area, called in the particular case of a waterplane, the centre of flotation (CF), from the y -axis is given by x ¯ = M y y A = ∫ x y 1 d x ∫ y 1 d x It is convenient to examine such a symmetrical figure in relation to the second moment of area, since it is normally possible to simplify work by choosing one symmetrical axis for ship shapes. The second moments of area or moments of inertia about the two axes for the waterplane shown in Fig. 2.15 are given by I T = 1 3 ∫ y 1 3 d x about O x for each half I y y = 1 3 ∫ x 2 y 1 d x about Oy for each half Fig. 2.15 The parallel axis theorem shows that the second moment of area of a plane figure about any axis, Q, of a set of parallel axes is least when that axis passes through the centre of area and that the second moment of area about any other axis, R, parallel to Q and at a distance h from it is given by (Fig. 2.16) I R = I Q + A h 2 Fig...
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...
