Area of Circles
The area of a circle is the measure of the space enclosed by the circle's boundary. It is calculated using the formula A = πr^2, where A represents the area and r is the radius of the circle. The concept is fundamental in geometry and has practical applications in various fields such as engineering, physics, and architecture.
6 Key excerpts on "Area of Circles"
eBook - ePub- Tandi Clausen-May(Author)
- 2013(Publication Date)
- SAGE Publications Ltd(Publisher)
...π is defined in terms of the circumference and the diameter – the distance around the circle, and the distance all the way across. But the standard formulas for the circumference and area of a circle, C = 2 π r, and A = π r 2, are expressed in terms of the circumference and the radius. This is the distance half way across the circle, not all the way. Learners need to understand clearly the difference between these two distances – which is easy if they are part of a picture, on paper or in the mind, but much more difficult if they are just squiggles on the page. So we can see – literally see if we draw or imagine a circle that just fits around a regular hexagon – that the circumference of a circle is 3 and a bit, or π, times the diameter. This is the same as π times double the radius, so C = π × 2r, or 2 π r. The formula can be taken directly out of the ‘picture in the mind’: there is no need to memorise it. PowerPoint PowerPoint 10-1, Circumference of a Circle, will help learners to establish the ‘picture in the mind’ that they need so that they can understand why the distance all the way round a circle is a little bit more than 3 times the distance straight across the middle. Theme: Mathematical Language – Radius, Diameter, Circumference These ‘mathematical’ terms may be associated with movements to make them more memorable. The learner stands and chants: I am the centre of the circle : Radius [Flings one arm straight out.] Diameter [Flings out the other arm.] Circumference ! [Turns right round on the spot.] Even if some learners are unwilling to take part in this activity they will still benefit from watching some of their classmates (and their teacher!) do it. b) The Area of a Circle Now that we have a way to visualise – to just see – that the circumference of a circle is π times the diameter, we are ready to think about its area. This needs a different ‘model to think with’...
eBook - ePubDyslexia, Dyscalculia and Mathematics
A practical guide
- Anne Henderson(Author)
- 2013(Publication Date)
- Routledge(Publisher)
...the: radius — r, diameter — D, circumference — C. Figure 9.16 Properties of a circle Circle facts ● The circumference is the perimeter of a circle. ● The diameter is twice the length of the radius. ● The radius is half the length of the diameter (divide D by 2). ● Pi π (pronounced pie) is important. ● Pi π has a value of 3.142. ● Press EXP on the calculator to use π. ● A 3D shape with circular top and bottom is a cylinder. To find the circumference of a circle: (answer is in units) π × D or π × 2r A rhyme to help: Fiddle de-dum, Fiddle de-dee The ring round the moon is π times D. Figure 9.17 How to find the circumference of a circle To find the area of a circle (answer is in units 2) π × radius × radius which is written πr 2 A rhyme to help: A round hole in my sock Has just been repaired. The area mended Is pi r squared. Figure 9.18 How to find the area (A) of a circle Polygons ● Copy, cut out, stick onto card and turn the angle pictures and facts given into a memory card. (number 23, see page 144). ● Multi-sided figures are generally called polygons. They have individual names depending on the number of sides, but many students find these difficult to remember. Figure 9.19 Polygons Section E: Co-ordinates The two straight lines at right angles to each other on a graph are called the axes. Coordinates are a pair of numbers, usually in brackets, which describe the precise location of a point on the axes. The one which is horizontal is called the x -axis (because x is a cross) and the vertical line is called the y -axis. The first number indicates the x -axis value (across the hall) and the second number indicates the y -axis value (up the stairs). For example: (3, 5) means 3 units across to the right and 5 units up. Figure 9.20 Graph to show the position of co-ordinate (3,5) Section F: Rotational symmetry This is the description given when a pattern is rotated around a point to identify the number of times the pattern is repeated...
- Alfred S. Posamentier, Stephen Krulik(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
...Using a toy dart set is preferred for classroom use rather than the archery set! Topic: Developing the Formula for the Area of a Circle Materials or Equipment Needed Construction paper prepared as shown in figures 9.1 and 9.2. Implementation of the Motivation Strategy Students are often “told” that the area of a circle is found by the formula A = π r 2. Too often, they are not given an opportunity to discover where this formula may have come from or how it relates to other concepts they have learned. It is not only entertaining, but also instructionally sound, to have the formula evolve from previously learned concepts. Assuming that the students are aware of the formula for finding the area of a parallelogram, this motivator presents a nice justification for the formula for the area of a circle. This motivational activity will use the teacher-prepared materials—a convenient size circle drawn on the piece of cardboard or construction paper, divided into 16 equal sectors (see figure 9.1). This may be done by marking off consecutive arcs of 22.5° or by consecutively dividing the circle into two parts, then four parts, then bisecting each of these quarter arcs, and so on. These sectors, shown in figure 9.1, are then to be cut apart and reassembled in the manner shown in figure 9.2. Figure 9.1 Figure 9.2 This placement suggests that we have a figure that approximates a parallelogram. That is, were the circle cut into more sectors, then the figure would approach a true parallelogram. Let us assume it is a parallelogram. In this case the base would have length, where C = 2π r (r is the radius). The area of the parallelogram is equal to the product of its base and altitude (which here is r). Therefore, the area of the, which is the commonly known formula for the area of a circle. This should certainly impress your students to the point where this area formula begins to have some intuitive meaning...
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...
- Rebecca Dayton(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...She has cut each of her 9-inch circular pies into eighths. What is the area of one slice of pie? Each slice of the pie is a sector of the circular pie. The diameter of the pie is 9 inches, so the radius is 4.5 inches and a slice represents of the pie. Therefore, the area of a pie slice is equal to square inches. Exercise 3 1. Find the length of. Round off your answer to the nearest hundredth. 2. Find the area of sector AOB. Equation of a Circle The standard form for the equation of a circle is used to define a circle on the coordinate plane. The equation is as follows: where h and k are the x - and y -coordinates of the center of the circle and r is the radius Example: Write the equation of a circle with center (−4, 0) and a diameter that measures 8 units. Since the center is located at (−4, 0), h = −4 and k = 0. The radius is half the length of the diameter, so the radius measures 4 units. Substituting into the standard form of the equation of the circle, (x − h) 2 + (y − k) 2 = r 2 (x − (−4)) 2 + (y − (0)) 2 = (4) 2 (x + 4) 2 + (y) 2 = 16 The equation of the circle is (x + 4) 2 + y 2 = 16. Example: Find the equation of the circle on the coordinate plane. Determine the coordinates of the center. (h, k) = (3, −1) Determine the length of the radius. r = 3 units Substitute into the standard form of the equation of the circle. (x − h) 2 +. (y − k) 2 = r 2 (x − 3) 2 + (y − (−1)) 2 = (3) 2 (x − 3) 2 + (y + 1) 2 = 9 The equation of the circle is (x − 3) 2 + (y + 1) 2 = 9. Example: Determine the center and the radius of the circle. (x + 2) 2 + y 2 = 49 The center is located at the point (h, k) given the formula (x − h) 2 + (y − k) 2 = r 2. Therefore, (x + 2) 2 + y 2 =...
eBook - ePub- Kylie A. Peppler(Author)
- 2017(Publication Date)
- SAGE Publications, Inc(Publisher)
...Maria Droujkova Maria Droujkova Droujkova, Maria Ray Droujkov Ray Droujkov Droujkov, Ray Math Circles Math circles 469 472 Math Circles A math circle is a social occasion to enjoy mathematics as a creative tool, a topic of philosophical discourse, or a game. A math circle is also a group of people who meet to study mathematics for its own sake, because they are interested. Math circles come in different types defined (a) by the target audience (e.g., children, teachers, or families), (b) by the format of activities (e.g., problem solving, projects, or discussions), and (c) by their organization (e.g., one-time gatherings, short-term cooperatives, and ongoing groups). Some math circles are informal gatherings of friends at someone’s home or a park, while others are run by organizations with dedicated staff and systematic planning. Math circles can be as small as three or four students or as large as hundreds of participants. Other entities similar to math circles include maker groups, robotics groups, science clubs, and coding communities. This entry discusses the roles and purpose of math circles, their prevalence and the different types that exist, the resources that are available for math circles, and the origin of math circles. Roles and Purpose of Math Circles What roles do math circles fill in the communities that support them, and what do they give their participants? It is common for math circle leaders to describe their circles in contrast to math classes in school, because math circles provide mathematical experiences that are different in some way, fulfilling a need not otherwise addressed. Two such differences are breadth and depth. The website of The Math Circle in Boston says, “We are careful to choose topics which are unlikely to be in the school curriculum—we see our role as widening and deepening the river, rather than accelerating its flow between narrow banks” (The Math Circle, n.d.)...
