12. Can you draw a polygon on the surface of a sphere? Give reasons for your answer. 13. Five children at play in a rectangular yard want to locate a point that is equally distant from the four corners. How can such a point be located? 96. The sum of the angles of a polygon. The diagonal of the quadrilateral, Figure 88, divides it into how many triangles? What is the sum of the angles of a triangle? Then what is the sum of the angles of the quadrilateral? The diagonals of the pentagon, Figure 89, divide it into how many triangles? What is the sum of the angles of this pentagon? Draw the Draw a hexagon. diagonals from one vertex. How many triangles are formed? What is the sum of the angles of a hexagon? The above examples show that if we draw all the diagonals possible T D A B FIG. 88. S R FIG. 89. from one vertex of a polygon, the polygon is divided into a number of triangles. This number is two less than the num ber of sides of the polygon. The sum of the angles of each Hence if S is the sum of the S= (n-2)180°. Exercise 88 1. If the angles of a hexagon are equal, how many degrees are there in each angle? 2. How many degrees in the sum of the angles of an octagon? 3. Four of the angles of a pentagon are 40°, 95°, 120°, and 100°. How many degrees in the other angle? 4. Three of the angles of a hexagon are equal. The other angles are 50°, 65°, and 80°. How many degrees in each of the three equal angles? 5. If the angles of a polygon of 12 sides are all equal, how many degrees in each angle? If the angles of a polygon of n sides are all equal, how many degrees in each angle? 6. Copy and fill out the following table : Name of polygon Number of sides Sum of the angles 97. Regular polygons. A polygon whose sides are all equal and whose angles are all equal is called a regular polygon. FIG. 90. FIG. 91. A B FIG. 92. A line from the center to the vertex of a regular polygon is called the radius of the polygon. Thus OA is the radius of the regular hexagon, Figure 92. The angle between two successive radii is called the central angle of the regular polygon. Angle AOB is the central angle of the regular hexagon. The perpendicular from the center to one side is called the apothem of the regular polygon. OC is the apothem of the regular hexagon. Exercise 89 1. What other name has a regular triangle? What other name has a regular quadrilateral? 2. How many degrees in each angle of a regular triangle? Of a regular quadrilateral? 3. How many degrees in the sum of the angles of a regular hexagon? Of a regular octagon? 4. How many degrees in each of the angles of a regular polygon of 5 sides? Of 6 sides? Of 8 sides? Of 10 sides? 5. How many central angles has a regular hexagon? How many degrees in the sum of all of them? In one of them? Answer the same questions for a regular pentagon; for a regular octagon; for a regular decagon. 6. How many equilateral triangles can be placed so that each has one vertex at a given point P, all of the triangles lying in the same plane? How many squares can be so placed? How many pentagons? How many hexagons? 7. If you were choosing tiles to cover a floor, and were required to use a single form of regular polygon, except along the edges, what forms might be chosen? 8. Find the perimeter of a regular hexagon one side of which is 10 in. 9. Make a formula for finding the perimeter, p, of a regular polygon of n sides, each side being s inches long. 10. A polygon is said to be inscribed in a circle when the vertices of the polygon are on the circle. If a circle is divided into three or more equal parts, and consecutive points of division are joined by straight lines, a regular polygon is inscribed in the circle. Inscribe a square in a circle. See page 160. 11. Inscribe a regular hexagon in a circle. See page 127. 12. Inscribe an equilateral triangle in a circle by joining alternate vertices of a regular hexagon. 13. Inscribe a regular octagon in a circle. If you have the vertices of a square that is inscribed in a circle, how can you find the vertices of the regular octagon? 14. What is the sum of the angles of a regular quadrilateral? How many degrees in each angle? Answer the same questions for a regular pentagon; a regular octagon ; a regular hexagon. 15. Make a table, similar to the table in exercise 6, page 175, for finding the number of degrees in an angle of a regular polygon of 3, 4, 5, 6, 7, 8, 9, and 10 sides. 16. What is the formula for finding the sum of the angles of a polygon of n sides? Make a formula for finding the number of degrees, A, in an angle of a regular polygon of n sides. 17. A circle may be divided into equal parts by drawing equal angles at the center. How many degrees must there be in each central angle to divide the circle into 3 equal parts? Into 4 equal parts? Into 5 equal parts? Into 8 equal parts? 18. Using the protractor to construct equal central angles, inscribe in a circle a regular pentagon; a regular octagon ; a regular decagon. 19. Tell how the two 8-pointed star-polygons on page 163 may be made by using the protractor. 20. Using the protractor, divide a circle into ten equal parts. By joining the points of division in different ways make three different kinds of 10-pointed star-polygons. Exercise 90. Review 1. Draw line segments a, b, and c and then construct a segment equal to 2 a-b+c. 2. Show how to obtain an angle by rotating a line. 3. Does changing the lengths of the sides of an angle change the size of the angle? Is an angle a portion of a plane surface? 4. If A 8° 41' 10", find its complement and its supple ment. 5. A man faces east. He turns to the left through 225°. In what direction is he then facing? Through how many degrees must he turn to the left to be facing east again? 6. Two angles of a triangle are equal and their sum is equal to the third angle. How many degrees in each angle? 7. Through how many degrees does a point on the circumference of a wheel turn while the wheel makes 4 revolutions? 8. Through how many degrees does a point on the surface of the earth turn in 1 hour? In 6 hours? In 1 minute? In 1 second? 9. Define parallel lines. 10. Define base, altitude, and diagonal of a parallelogram. 11. Define a regular polygon. 12. State the five principles concerning quadrilaterals given on page 171. 13. Construct the pattern in Figure 93. 14. Figure 94 is used as the basis of Gothic windows. Make it. D is the mid-point of AC. The center of the circle is found by drawing arcs with A and B as centers and BD as a radius. |