POLYGONS
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1 POLYGONS After studying triangles and quadrilaterals, students now extend their study to all polygons, with particular attention to regular polygons, which are equilateral and equiangular. Using the fact that the sum of the measures of the angles in a triangle is, students describe a method to determine the sum of the measures of the interior angles of any polygon. Next they explore the sum of the measures of the exterior angles of a polygon. Finally they use the information about the angles of polygons along with their triangle tools to determine the angle measures and areas of regular polygons. For additional information see the Math Notes boxes in Lessons and Example 1 The figure at right is a hexagon. What is the sum of the measures of the interior angles of a hexagon? Explain how you know. Then write an equation and solve for x. + 7º + 1º x + 1º 2x 5º 4º One way to calculate the sum of the interior angles of the hexagon is to divide the polygon into triangles. One way to divide the hexagon into triangles is to draw all of the diagonals from a single vertex, as shown at right. Doing this forms four triangles, each with angle measures summing to m 1+ m 2 + m 3 + m 4 + m 5 + m 6 + m 7 + m 8 + m 9 + m 10 + m 11+ m 12 = 4() = CPM Educational Program. All rights reserved. Core Connections Integrated II Note: Students may notice that the number of triangles drawn from a single vertex is always two less than the number of sides. This example illustrates why the sum of the interior angles of a polygon may be calculated using the formula sum of interior angles = 180º(n 2), where n is the number of sides of the polygon. Now using the sum of the angles, write an equation, and solve for x. ( + 1º) + ( + 7º) + (x + 1º) + ( 5º) + ( 4º) + (2x) = 720º 18x = 720º x = 40º 2
2 Example 2 If the sum of the measures of the interior angles of a polygon is 2340, how many sides does the polygon have? Use the formula sum of interior angles = 180º (n 2) to write an equation and solve for n. The solution is shown at right. Since n = 15, the polygon has 15 sides. 180º(n 2) = 2340º 180ºn 360º = 2340º 180ºn = 2700º n = 15 It is important to note that if the answer is not a whole number, then either an error was made or there is no polygon with interior angles that sum to the given measure. Since the answer is the number of sides, the answer must be a whole number. Polygons cannot have 7.2 sides! Example 3 What is the measure of an exterior angle of a regular decagon? A decagon is a 10-sided polygon. The sum of the measures of the exterior angles of any polygon, one at each vertex, is always 360, no matter how many sides the polygon has. In this case the ten exterior angles are congruent since the decagon is regular. Therefore, each angle measures = 36. Example 4 A regular dodecagon (12-sided polygon) has a side length of 8 cm. What is the area of the dodecagon? Imagine dividing the dodecagon into twelve congruent triangles, radiating from the center, as shown at right. If we determine the area of one of the triangles, then we can multiply it by twelve to get the area of the entire dodecagon. One of the triangles is enlarged at right. The triangle is isosceles, so drawing a segment from the vertex angle perpendicular to the base is the height. Because the triangle is isosceles, the height bisects the base. h 8 Calculate the sum of all the interior angles of the dodecagon by using the formula () (12 2) = Parent Guide with Extra Practice 2015 CPM Educational Program. All rights reserved. 91
3 Since it is a regular dodecagon, all the interior angles are congruent, so each angle measures = 150. The segments radiating from the center form congruent isosceles triangles, so the base angles of each triangle measure 75 (half of the 150º angle, as shown in the diagram on the previous page). Use trigonometry to calculate the value of h as shown as right. It is best to use an unrounded value of h to calculate the area, and then round the answer appropriately at the end of your calculations. tan 75 = h 4 h = 4 tan 75 h cm Therefore the area of one of these triangles is: A 1 2 (8 cm)( cm) cm2 Multiply the area of one triangle by 12 to get the area of the entire dodecagon. A 12( cm) , or about 717 cm 2 Problems Calculate the measures of the angles in each problem below. 1. The sum of the interior angles 2. The sum of the interior angles of a heptagon (7-gon). of an octagon (8-gon). 3. The measure of each interior 4. The measure of each interior angle of a regular dodecagon (12-gon). angle of a regular 15-gon. 5. The measure of each exterior 6. The measure of each exterior angle of a regular 17-gon. angle of a regular 21-gon. Solve for x in each of the figures below x x x 2x Complete each of the following problems. 11. Each exterior angle of a regular n-gon measures How many sides does this n-gon have? 12. Each exterior angle of a regular n-gon measures How many sides does this n-gon have? 13. Each interior angle of a regular n-gon measures 156º. How many sides does this n-gon have? 14. Each interior angle of a regular n-gon measures 165.6º. How many sides does this n-gon have? CPM Educational Program. All rights reserved. Core Connections Integrated II
4 15. What is the area of a regular pentagon with side length 8.0 cm? 16. Calculate the area of a regular hexagon with side length 10.0 ft. 17. Calculate the area of a regular octagon with side length 12.0 m. 18. What is the area of a regular decagon with side length 14.0 in? + 6º 2º 19. Using the pentagon at right, write an equation and solve for x. x 1º 20. What is the sum of the measures of the interior angles of a 14-sided polygon? 21. What is the measure of each interior angle of a regular 16-sided polygon? 22. What is the sum of the measures of the exterior angles of a decagon (10-gon)? 23. Each exterior angle of a regular polygon measures How many sides does the polygon have? + 1º 6x + 3º 24. Is there a polygon with interior angle measures that add up to 3060? 25. Is there a polygon with interior angle measures that add up to 1350? 26. Is there a polygon with interior angle measures that add up to 4410? A 27. In the figure at right, ABCDE is a regular pentagon. Is EB DF? Justify your answer. E 72 B 28. What is the area of a regular pentagon with a side length of 10 units? 29. What is the area of a regular 15-gon with a side length of 5 units? D C F Parent Guide with Extra Practice 2015 CPM Educational Program. All rights reserved. 93
5 Answers º º º º º º 7. x = 24º 8. x = 30º 9. x 98.2º 10. x 31.3º sides sides sides sides cm ft m in x + 7º = 540º; x 28.1º sides sides 25. No. The result is not a whole number. 26. No. The result is not a whole number. 27. Yes. Since ABCDE is a regular pentagon, the measure of each interior angle is 108. Therefore, m DCB = 108º. Since DCB and FCB are supplementary, m FCB = 72. Thus EB DF because alternate interior angles FCB and EBC are congruent sq. units sq. units CPM Educational Program. All rights reserved. Core Connections Integrated II
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