1. Obj. 45 Circles & Polygons The student is able to (I can): Develop and use formulas to find the areas of circles and regular polygons Develop and use formulas to find arc length and sector area of circles

2. circleThe set of all points in a plane that are a fixed distance from a point, called the center. A circle is named by the symbol center. AAand its 3. diameterA line segment whose endpoints are on the circle and includes the center of the circle.radiusA line segment which has one endpoint on the circle and the other on the center of the circle. BC ADCD is a diameter AB is a radius 4. (pi)The ratio of the circumference to the diameter. C = d which becomes C = dSince the diameter is twice the radius, this formula can also be written as: C = 2r is an irrational number it never repeats and it never ends. The symbol is an exact number; 3.1415926 is an approximation. 5. If you cut a circle into wedges, and arrange the wedges into a parallelogram-shaped figure: A = bh 1 A = circumference i radius 2 1 A = ( 2radius ) i radius 22 A = rradius 1 2circumference 6. Examples1. Find the exact circumference and area of a circle whose diameter is 18. C = d = (18) = 18 A = r2 = (92) = 81 2. Find the diameter and area of a circle whose circumference is 22. 22 = d d = 22 A = (112) = 121 3. Find the radius of a circle whose area is 81. 81 = r2 81 = r2 r=9 7. central angle of a circleAn angle whose vertex is on the center of the circle, and whose sides intersect the circle.sector of a circleA region bounded by a central angle. RA GRAG is a central angle RAG is a sector 8. The area of a sector is proportional to the area of the circle containing the sector. Area of sector central angle = Area of circle 360 S m = 2 r 360Formula: m S = r 360 2 9. ExamplesFind the area of each sector. Leave answers in terms of . 120 S = (2 ) 360 4 i120 = 360 4 = in.2 3 2 72 S = ( 10 ) 360 7200 = 360 21. 120 2" 2. 72 10m= 20 m2 10. chordA line segment whose endpoints are on the circle.arcA portion of a circle.KKY is a chord. EYKY is an arc. 11. arc measureThe measure of the central angle that intercepts the arc.arc lengthThe distance along an arc. It is proportional to the circumference of the circle. arc length central angle = circumference 360 m m m L = C 360 where C is the circumference (either C=d or C=2r). 12. ExampleFind each exact arc length. 1.120 32.8m 72120 L = 2 ( 3 ) 360 = 2 ft.72 L = 2 ( 8 ) 360 =16 or 3.2 m 5 13. apothemThe distance from the center of a polygon to one of the sides.central angleAn angle whose vertex is at the center and whose sides go through consecutive vertices.central angle 14. For regular polygons, likewise, we can cut out a regular polygon and arrange the halves as shown: A = bh 1 A = perimeter i apothem 2 1 A = aP 2apothem1 2perimeter 15. Since all of the central angles have to add up to 360, the measure of a central angle is 360 n central angleapothem 16. If we look at the isosceles triangle created by a central angle (the apothem is the height), one of the triangles base angle is the interior angle. central angleinterior angleapothem interior angle 17. What this means is that on any regular polygon, we can set up a right triangle, with one leg as the apothem and one leg as half the length of a side. central angleapothem interior angle sideWhen you are calculating areas, you will almost always be using the tangent ratio (opposite/adjacent). 18. Lets review our table of interior angles and calculate what the values will be for the angles of their right triangles: SidesNameEach Int. Int. Central Central 3Triangle6030120604Quadrilateral904590455Pentagon1085472366Hexagon1206060307Heptagon128.664.351.425.78Octagon13567.54522.59Nonagon14070402010Decagon14472361812Dodecagon15075nn-gon30 360 n15 180 n(n 2 ) 180 (n 2 ) 90 nn 19. Examples1. Find the area of the regular pentagon: 10Lets sketch in our right triangle:36 a 54 5To find the apothem (a), we can use the tangent ratio: 5 a tan 36 = tan54 = or 5 a Thus, a 6.88. The perimeter = 50, so 1 A = (6.88)(50) = 172 2 20. Examples2. Find the area of the equilateral triangle:12 60 2 36This is a 30-60-90 triangle, so the apothem is going to be 6 3 = 2 3 3 Since it is equilateral, the perimeter will be 3(12) = 36. So the area is 1 1 A = aP = 2 3 ( 36 ) = 36 3 2 2() 21. Because the central angle of an equilateral triangle forms a special right triangle, we also have a special formula for the area of an equilateral triangle: s2 3 (Where s is the A= side length.) 4 Looking at the previous example, if we plug in 12 for s, we get: 122 3 144 3 A= = = 36 3 4 4 The advantage of this is that we dont have to try to draw in that tiny 30-60-90 triangle. Remember this as s-2-3-4. 22. Likewise, because the hexagon is composed of 6 equilateral triangles, we have a special formula for the area of a hexagon: s2 3 A = 6 4 Example: What is the area of a hexagon with a side length of 14? 14 2 3 196 3 A = 6 = 6 4 4 ()= 6 49 3 = 294 3 23. Summary Circle formulas C = d or C = 2r A = r2 Regular polygon formulas 1 A = aP 2 s2 3 Equilateral triangle: A = 4 s2 3 Regular hexagon: A = 6 4