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Geometry

Learn how to find the area of equilateral triangles and use apothem to calculate the area of regular polygons. Includes quick review of 30-60-90 triangles and right triangle trigonometry.

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Geometry

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  1. Geometry Areas of Regular Polygons

  2. Goals • Find the area of equilateral triangles. • Know what an apothem is and be able to find its length. • Use the apothem to find the area of a regular polygon.

  3. Quick Review • 30-60-90 Triangles • Right Triangle Trigonometry • Area of a triangle

  4. 30-60-90 Triangle 30 2a 60 a

  5. Trig Ratio Definition: Tangent Hypotenuse Opposite A Adjacent Opposite Tangent of A = Adjacent

  6. Area of any Triangle h b

  7. 11.2 Area of Regular Polygons

  8. s s s Area of an Equilateral Triangle h ? base

  9. s s s Finding h. We can solve for h by using the Pythagorean Theorem. h ?

  10. Finding h. s s h

  11. Solving for Area s s s

  12. Area of an Equilateral Triangle s s s

  13. Example Find the area. Solution: 8 8 16 8

  14. Your Turn Find the area. 10 10 25 10

  15. 5.89 5.89 5.89 Example 2 The area of an equilateral triangle is 15. Find the length of the sides.

  16. Area of a Regular Hexagon Divide the hexagon into six equilateral triangles. Each triangle has an area of s

  17. Area of a Regular Hexagon Multiply this by 6: s

  18. Find the area of a regular hexagon with side length of 8. Example 32 8

  19. Segments in a regular polygon. Center Radius Apothem

  20. Apothem • The perpendicular distance from the center of a regular polygon to one of its sides is called the apothem or short radius. It is the same as the radius of a circle inscribed in the polygon. • Apothem is pronounced with the emphasis on the first syllable with the a pronounced as in apple (A-puh-thum).

  21. Apothem Radius Apothem

  22. Another Way to Find the Area The area of the hexagon is equal to the area of one triangle multiplied by the number of triangles, n. Area = (Area of one )  (Number of s)

  23. Area of one triangle Radius r Apothem a This is the Area of only one triangle. s

  24. Area of one triangle Remember, there are n triangles. The total area then is r a s

  25. Perimeter The perimeter of the hexagon is s  n. p = s  n s s s r s s a s

  26. Area of a Regular Polygon This formula works for all regular polygons regardless of the number of sides. a = apothem p = perimeter

  27. Example Find the area. • Draw a radius and an apothem. • What kind of triangle is formed? • 30-60-90 • What is the length of the segment marked x? • 6 a r 12 60 6 x

  28. Example Find the area. 4. So what is r? 12 5. And what is a? 6. The perimeter is? 72 (6  12) a 12 r 12 6

  29. Example Find the area. The apothem is and the perimeter is 72. The area is 12 12

  30. Universal Formula Click Here to Skip

  31. Another Very Useful Formula • Given the length of a side, s, of a regular polygon with n sides: • n = the number of sides • s = the length of a side

  32. Previous Example Again 12 (graphing calculator)

  33. Notice! • In a regular hexagon, the radius is always equal to the length of a side. • This is because we divide the hexagon into equilateral triangles. • A hexagon is the onlyshape where this is true.

  34. The Fly in the Ointment… • If the polygon is anything other than an equilateral triangle, a square, or a regular hexagon, finding the apothem and the radius can be very challenging. • Use what you know about 30-60-90 triangles, 45-45-90 triangles, and even trig to solve the problem.

  35. A harder example Find the area of the regular pentagon. Where did 36 come from? Each central angle measures 1/5 of 360, or 72. The apothem bisects the central angle. Half of 72 is 36. 360 36 6

  36. A harder example Find the area of the regular pentagon. What is the apothem? 6 What is the perimeter? Don’t know. Let’s find it. 36 6

  37. A harder example Find the area of the regular pentagon. What trig function can be used to find x? TANGENT (SOHCAHTOA) Equation: 36 6 x

  38. A harder example Solve the equation: 36 6 x Use a scientific calculator or use the table on page 845.

  39. A harder example x = 4.36 One side of the pentagon measures? 8.72 (2  4.36) The perimeter is 43.59 (5  8.72) 36 8.72 6 4.36

  40. A harder example The area is: 36 8.72 6 x

  41. Final Example Find the area of a regular octagon if the length of the sides is 10.

  42. Step 1 • Draw a regular octagon with side length 10. 10

  43. Step 2 • Locate the center and draw a central angle. 10

  44. Step 3 • Determine the measure of the central angle. 10 45

  45. Step 4 • Draw the apothem. 10 45

  46. Step 5 • The apothem bisects the angle and the side. Write their measures. 10 22.5 45 5

  47. Step 6 • Use a trig function to find the apothem. 10 22.5 a 5

  48. Step 7 • Find the perimeter. p = 10  8 p = 80 10 12.07

  49. Step 8 • Find the area. p = 80 A = 482.8 10 12.07

  50. Using the area formula: Skip This 10

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