Area of Regular Polygons
The area of a regular polygon is the measure of the space enclosed by the polygon's sides. To find the area, you can use the formula A = 1/2 * apothem * perimeter, where the apothem is the distance from the center to a side and the perimeter is the total length of the sides. This formula applies to all regular polygons, such as squares, hexagons, and octagons.
5 Key excerpts on "Area of Regular Polygons"
- Rebecca Dayton(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Concave A polygon is a two-dimensional closed shape with straight lines and is either convex or concave. Convex polygons are polygons whose interior angles are each less than 180°. That is, no angles point inward. Any polygon that is not convex is considered a concave polygon. Regular vs. Irregular A regular polygon is a polygon with all equal angles and sides. If a polygon does not have equal angles and sides, then it is an irregular polygon. Interior Angles of Polygons The sum of the interior angles of a polygon can be found using the following formula: (n − 2)180° where n is the number of sides in the polygon. Example: Find the sum of the interior angles of a 12-sided polygon. (n − 2)180° (12 − 2)180° (10)180° 1800° The sum of the interior angles of a 12-sided figure is 1800°. Example: Find the measure of each unknown angle, x. First, find the sum of the interior angles of the polygon. Since the polygon has 5 sides, it is a pentagon. The sum of the interior angles is determined as follows: (n − 2)180° (5 − 2)180° (3)180° 540° The sum of the interior angles of the figure is 540°. Using this sum, the following equation can be used to find the value of x : Exterior Angles of Polygons Regardless of the number of sides in the polygon, the sum of the exterior angles is always 360°. An interior angle and one of the exterior angles at the same vertex are supplementary. Example: Steven is drawing a regular octagon. What is the measure of one exterior angle of the octagon? The sum of the exterior angles of any polygon is 360°. Since Steven is drawing an octagon, the figure will have 8 exterior angles. Each exterior angle is congruent because the octagon is regular...
eBook - ePub- Sandra Lee(Author)
- 2020(Publication Date)
- Wiley-Blackwell(Publisher)
...Appendix Mathematical Formulae and Applied Mensuration Formulae for areas (A) of plane figures or (Circumference = π × D or 2πr) Where S = area of sector T = area of triangle or approximately Where H = rise C = chord Regular polygons Pentagon (5 sides) A = S × S × 1.720 Hexagon (6 sides) Heptagon (7 sides) Octagon (8 sides) Nonagon (9 sides) Decagon (10 sides) Irregular figures Divide figure into trapezoids by equidistant parallel lines (ordinates or offsets) Where S = distance between ordinates P = sum of first and last ordinates Q = sum of intermediate ordinates or Simpson's rule (must be even number of trapezoids) Where Z = sum of even intermediate ordinates Y = sum of odd intermediate ordinates Formulae for surface areas (SA) and volume (V) of solids Where B = area of base (π × r × r) C = circumference (π × D) Where B = area of base (π × r × r) C = circumference of base SH = slope. height Where B = area of base b = area of top R = radius at base r = radius at top SH = slope height Where B = area of base P = perimeter of base SH = slope height Where B = area of base b = area of top P = perimeter of base p = perimeter of top SH = slope height Where R = radius at base r = radius at top Where R = radius of base Irregular areas Any irregular‐shaped area to be measured is usually best divided up into triangles, the triangles being measured individually and then added to give the area of the whole – if one of the sides, as for instance in the case of paving, is irregular or curved, the area can still be divided into triangles by the use of a compensating, or give and take, line, i.e. a line is drawn along the irregular or curved boundary in such a position that, so far as can be judged, the area of paving excluded by this line is equal to the area included beyond the boundary...
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 - ePub- Tandi Clausen-May(Author)
- 2013(Publication Date)
- SAGE Publications Ltd(Publisher)
...C HAPTER 9 Perimeter, Area and Volume Some key concepts A perimeter is a distance. An area may be thought of as ‘an amount of flatness’. A parallelogram can be sheared back into a rectangle with the same area. A triangle can be sheared back into a right-angled triangle with the same area. A container has a capacity. A solid has a volume. A cubic container measuring 10cm by 10cm by 10cm has a capacity of 1 litre. It will hold 1 litre of water, which weighs 1 kilogram. a) Vocabulary School mathematics is steeped in hard words. Indeed, language is a theme that crops up repeatedly throughout this book. But nowhere is it more evident than in geometry and measures. Mathematical language may present a major hurdle to learners who could otherwise fly with the ideas and images of shape and space, causing them instead to crash on the mass of mathematical jargon. Kilogram, perimeter, pentagon … these are all hard words, although they refer to quite straightforward ideas. It is worth spending time making as much sense as possible out of the jargon, demystifying it wherever you can. Pentagon, for example, is simply the Greek for five sides. Talking in Greek is no more mathematically correct than talking in English – so pentagon is not a more mathematical term than five sides. Our use of Greek and Latin in the mathematics classroom is just a historical accident – and it is not helpful to learners who may find it harder to remember the new words than to understand what they mean in the first place. Learners need to acquire a lot of mathematical jargon in order to achieve under our curriculum and assessment structure. However, it is important always to keep a clear distinction between the mathematical concepts that need to be understood, and the vocabulary used to describe them. So, for example, learners may identify pairs of shapes that are exactly the same, and others that are the same shape but different sizes, long before they learn the ‘mathematical’ terms congruent and similar...
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...
