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11.1 – Angle Measures in Polygons

11.1 – Angle Measures in Polygons. Diagonals  Connect two nonconsecutive vertices, and are drawn with a red dashed line. Let’s draw all the diagonals from 1 vertex. Sides # of Triangles Total degrees. 3. 540. 5. Find out how many degrees are in these two shapes, and try to make a formula.

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11.1 – Angle Measures in Polygons

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  1. 11.1 – Angle Measures in Polygons

  2. Diagonals  Connect two nonconsecutive vertices, and are drawn with a red dashed line. Let’s draw all the diagonals from 1 vertex. Sides # of Triangles Total degrees 3 540 5

  3. Find out how many degrees are in these two shapes, and try to make a formula Sides # of Triangles Total degrees 540 3 5 720 4 6 5 900 7 n-2 (n-2)180 n

  4. Remember, angles on the outside are EXTERIOR ANGLES. What do all the exterior angles of a octagon add up to? What do all the Exterior Angles of a polygon add up to? What do all the exterior angles of a decagon add up to? 360 degrees!!

  5. Theorem 11-1 (Sum of interior angles of polygon)  The sum of the measures of the angles of a convex polygon with n sides is (n-2)180 Theorem 11-2 (Exterior angles sum theorem)  The sum of the measure of the exterior angles of a convex polygon is 360.

  6. What is the measure of one interior angle of a regular pentagon? What is the measure of one interior angle of a regular octagon? The general formula for the measure of one interior angle of a REGULAR polygon is 

  7. Fill out this regular polygon chart here. Think about the relationship between interior and exterior angles. Interior and exterior angles are supplementary.

  8. Sum of interior angles in polygon Sum of exterior angles in polygon Measure of ONE interior angle of REGULAR polygon Measure of ONE exterior angle of REGULAR polygon

  9. How many sides are there if the one interior angle of a regular polygon is 135 degrees? How many sides are there if the one exterior angle of a regular polygon is 45 degrees? How many sides are there if the one interior angle of a regular polygon is 170 degrees? How many sides are there if the one exterior angle of a regular polygon is 20 degrees? Interior and exterior angles are supplementary.

  10. 11.2 – Areas of Regular Polygons

  11. Area of Equilateral triangle. s 8

  12. Central Angle  Angle formed from center of polygon to consecutive vertices. Radius Apothem  Distance from center of polygon to side. Things to notice, all parts can be found using SOHCAHTOA. It is isosceles, so you can break up the triangle in half.

  13. The area of these 5 triangles is = Or we can think of it as What do you think we can do to find the area of this shape? So you see it’s

  14. Let’s find the area of a pentagon with side length 10 Which trig function do we use to find the apothem? 72o TANGENT! Plug in, be careful with the perimeter! 36o 5 10

  15. 10 10

  16. 11.3 – Perimeters and Areas of Similar Figures

  17. Find the perimeter and area of a rectangle with dimensions: 4 by 10 28 40 8 by 20 56 160 6 by 15 42 90 20 by 50 140 1000 2 by 5 14 10 Side RatioPerimeter RatioArea Ratio 1:5 1:25 1:5 4:3 16:9 4:3 3:1 9:1 3:1

  18. Do you notice a relationship between the side ratio, perimeter ratio, and area ratio? Theorem 11-5 If the scale factor of two similar figures is a:b, then: 1) The ratio of perimeters is a:b 2) The ratio of areas is a2:b2 Find the area and perimeter of a rectangle with dimensions: 4 by 10 28 40 8 by 20 56 160 6 by 15 42 60 20 by 50 140 1000 2 by 5 14 10 Side RatioPerimeter RatioArea Ratio 1:5 1:25 1:5 4:3 16:9 4:3 3:1 9:1 3:1

  19. Find the perimeter ratio and the area ratio of the two similar figures given below.

  20. Two basic problems: I have two pentagons. If the area of the smaller pentagon is 100, and they have a 1:4 side length ratio, then what is the area of the other pentagon? I have 2 dodecagons. If the area of one is 314 and the other is 942, what is the side length ratio?

  21. Two basic problems: A cracker has a perimeter of 10 inches. A similar mini cracker has perimeter 5 inches. If the area of the regular cracker is 20 in2, what is the area of the mini cracker? I have 2 n-gons. If the area of one is 135 and the other is 16, what is the perimeter ratio?

  22. 11.4 – Circumference and Arc Length

  23. Circumference is the distance around the circle. (Like perimeter) C = πd = 2πr LIKE THE CRUST PIZZA PART Area of a circle: A = πr2

  24. A O x B Like crust

  25. Find the length of the arc 3 O 120o

  26. Find the length of the arc 5 O 100o

  27. Find the length of the arc O 30 20o

  28. Find x and y O

  29. Find the Perimeter of this figure. 12 20 Do not subtract and then square, must do each circle separately!

  30. Find Perimeter of red region. 4

  31. Find the length of green part 30o 6

  32. 11.5 – Areas of Circles and Sectors

  33. Circumference is the distance around the circle. (Like perimeter) C = πd = 2πr LIKE THE CRUST PIZZA PART Area of a circle: A = πr2

  34. Find the area of a circle with diameter 8 in.

  35. Fake sun has a radius of .5 centimeters. Find the circumference and area of fake sun. Area: π(.5)2 = .25π Circumference: 2π(.5) = π

  36. Find the area of the shaded part. 10 8 5 6

  37. A O x B Like crust Like the slice

  38. Find the area of the sector. 3 O 120o

  39. Find the area of the sector. 4 O 90o

  40. Find the area of the sector. 10 O 160o

  41. Find area of blue part and length of green part 30o 6

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