1. Areas of Regular Polygons
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.
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3. Quick Review 30-60-90 Triangles Right Triangle Trigonometry
Area of a triangle
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4. Triangle
30 2a
60 a
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5. Trig Ratio Definition: Tangent
Hypotenuse
Opposite A
Adjacent
Opposite
Tangent of A =
Adjacent
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6. Area of any Triangle h b
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7. 11.2
Area of
Regular
Polygons
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8. Area of an Equilateral Triangles h ?
base
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9. Finding h. We can solve for h by using the Pythagorean Theorem. s h ?
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10. Finding h. s s h
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11. Solving for Area s s s
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12. Area of an Equilateral Triangles s s
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13. Example Find the area.
Solution: 8 8 16 8
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14. Your Turn Find the area. 10 10 25 10
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15. Example 2
The area of an equilateral triangle is 15. Find the length of the sides.
5.89
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16. Area of a Regular Hexagon
Divide the hexagon into six equilateral triangles.
Each triangle has an area of s
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17. Area of a Regular Hexagon
Multiply this by 6: s
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18. Example 32 8 Find the area of a regular hexagon with side length of 8.
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19. Segments in a regular polygon.
Center
Radius
Apothem
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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).
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21. Apothem
Radius
Apothem
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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)
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23. Area of one triangle r a s Radius Apothem
This is the Area of only one triangle. s
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24. Area of one triangle r a s Remember, there are n triangles.
The total area then is r a s
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25. Perimeter The perimeter of the hexagon is s  n. p = s  n s s s r s s
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26. Area of a Regular Polygon
This formula works for all regular polygons regardless of the number of sides.
a = apothem
p = perimeter
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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 r a 12
60 x 6
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28. Example Find the area. 4. So what is r? 12 5. And what is a?
6. The perimeter is?
72 (6  12) 12 r a 12 6
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29. Example Find the area. The apothem is and the perimeter is 72.
The area is 12 12
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30. Universal Formula
Click Here to
Skip
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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
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32. Previous Example Again12
(graphing calculator)
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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 only shape where this is true.
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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 triangles, triangles, and even trig to solve the problem.
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35. A harder example 360 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.
36
360 6
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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
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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
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38. A harder example Solve the equation: 36 6 x
Use a scientific calculator or use the table on page 845.
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39. A harder example x = 4.36 One side of the pentagon measures?
8.72 (2  4.36)
The perimeter is
(5  8.72)
36
8.72 6
4.36
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40. A harder example
The area is:
36
8.72 6 x
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41. Final Example
Find the area of a regular octagon if the length of the sides is 10.
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42. Step 1
Draw a regular octagon with side length 10. 10
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43. Step 2
Locate the center and draw a central angle. 10
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44. Step 3 Determine the measure of the central angle. 10 45
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45. Step 4
Draw the apothem. 10
45
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46. Step 5
The apothem bisects the angle and the side. Write their measures. 10
22.5
45 5
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47. Step 6 Use a trig function to find the apothem. 10 22.5 a 5
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48. Step 7
Find the perimeter.
p = 10  8
p = 80 10
12.07
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49. Step 8
Find the area.
p = 80
A = 482.8 10
12.07
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50. Using the area formula:
Skip
This 10
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51. Summary
The area of any regular polygon can be found be dividing the shape into congruent triangles, finding the area of one triangle, then multiplying by the number of triangles.
Or, multiply the length of the apothem by the perimeter and divide that by 2.
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52. Practice Problems
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