1. side length: triangle:
11.3B Warm Up
1. Find the measure 2. Find the missing 3. Find the mising
of c and b: side lengths: side lengths:
4. Find the missing Find the height of the
side length: triangle:

2. 11.3 Areas of Regular Polygons
Geometry
11.3 Areas of Regular Polygons

3. 11.3 Areas of Regular Polygons
Essential Question
How can you find the area of a regular polygon?
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11.3 Areas of Regular Polygons

4. 11.3 Areas of Regular Polygons
Goals
Find the angle measures in regular polygons
Find areas of regular polygons
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11.3 Areas of Regular Polygons

5. 11.3 Areas of Regular Polygons
Quick Review
Triangles
Right Triangle Trigonometry
Area of a triangle
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11.3 Areas of Regular Polygons

6. 11.3 Areas of Regular Polygons
Triangle
30 2a
60 a
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11.3 Areas of Regular Polygons

7. 11.3 Areas of Regular Polygons
Area of
Regular
Polygons
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11.3 Areas of Regular Polygons

8. Area of an Equilateral Triangles
30°
30° ? h
60°
60°
base
Remember: An equilateral triangle is also equiangular.
So… all angles are ___°.
60°
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11.3 Areas of Regular Polygons

9. 11.3 Areas of Regular Polygons
Finding h. s
We can solve for h by using the Triangle. h
a 3
1 2 𝑠 3 =
3 2 𝑠
60°
60°
𝑎= 1 2 𝑠 ?
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11.3 Areas of Regular Polygons

10. 11.3 Areas of Regular Polygons
Solving for Area s s s
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11.3 Areas of Regular Polygons

11. Area of an Equilateral Triangles s s
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11.3 Areas of Regular Polygons

12. 11.3 Areas of Regular Polygons
Example Find the area.
Solution:
𝐴= 1 2 𝑏ℎ
𝐴= 1 2 ∙8∙4 3
𝐴=16 3 8 8 4
𝟒 𝟑 4 8
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11.3 Areas of Regular Polygons

13. 11.3 Areas of Regular Polygons
Your Turn Find the area.
𝐴= 1 2 𝑏ℎ
𝐴= 1 2 ∙10∙5 3
𝐴=25 3 10 10 5 10
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11.3 Areas of Regular Polygons

14. 11.3 Areas of Regular Polygons
Example 2
𝐴= 𝑠 2
15= 𝑠 2
𝑠 2 = ≈34.64
𝑠=5.89
The area of an equilateral triangle is 15. Find the length of the sides.
5.89
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11.3 Areas of Regular Polygons

15. Segments in a regular polygon.
A Central Angle is formed by two radii to consecutive vertices.
Center
Central
angle
Radius
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11.3 Areas of Regular Polygons

16. Segments in a regular polygon.
Center
Radius r
Apothem a
Side s
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11.3 Areas of Regular Polygons

17. 11.3 Areas of Regular Polygons
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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11.3 Areas of Regular Polygons

18. 11.3 Areas of Regular Polygons
Apothem
Radius
Short
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11.3 Areas of Regular Polygons

19. Segments in a regular polygon.
Measure of the Central Angle of a regular polygon with n sides:
𝑚∠1= 360 𝑛
Center
Radius
Central
angle
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11.3 Areas of Regular Polygons

20. Measure of a central angle in a regular polygon.
Central Angle of a Hexagon:
x= 360 𝑛
x= 360 6
x=60° x
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11.3 Areas of Regular Polygons

21. Segments in a regular polygon.
The radii are congruent.
What are the measures of the other two angles? ____
This is an equilateral Δ.
60°
60°
60°
60°
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11.3 Areas of Regular Polygons

22. Area of a Regular Hexagon
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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11.3 Areas of Regular Polygons

23. 11.3 Areas of Regular Polygons
Area of one triangle
Radius r
Apothem a
This is the Area of only one triangle. s
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11.3 Areas of Regular Polygons

24. Area of a regular polygon
Remember, there are n triangles.
The total area then is r a s
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11.3 Areas of Regular Polygons

25. 11.3 Areas of Regular Polygons
Perimeter
The perimeter of the hexagon is s  n.
p = s  n s s s r s s a s
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11.3 Areas of Regular Polygons

26. Area of a Regular Polygon
𝐴= 1 2 𝑎𝑠𝑛 or 𝐴= 1 2 𝑎𝑝
a = apothem
s = side length
n= number of sides
p = perimeter
This formula works for all regular polygons regardless of the number of sides.
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11.3 Areas of Regular Polygons

27. 11.3 Areas of Regular Polygons
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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11.3 Areas of Regular Polygons

28. 11.3 Areas of Regular Polygons
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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11.3 Areas of Regular Polygons

29. 11.3 Areas of Regular Polygons
Example Find the area.
The apothem is
and the perimeter is 72.
The area is 12 12
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11.3 Areas of Regular Polygons

30. 11.3 Areas of Regular Polygons
Example
Find the area of a regular hexagon with side length of 8.
𝐴= 1 2 𝑎𝑠𝑛
𝐴= 1 2 ∙4 3 ∙8∙6
𝐴=96 3 2
4 3 4 8
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11.3 Areas of Regular Polygons

31. 11.3 Areas of Regular Polygons
Your Turn Find the area.
𝐴= 1 2 𝑎𝑠𝑛 9
𝐴= 1 2 ∙9 3 ∙18∙6
9 3
𝐴=486 3 18
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11.3 Areas of Regular Polygons

32. 11.3 Areas of Regular Polygons
Example Find the area.
𝐴= 1 2 𝑎𝑠𝑛 3
𝐴= 1 2 ∙6∙4 3 ∙6
x 3 =6 6
x= 6 3 ∙
𝐴=72 3
4 3 x=
x=2 3
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11.3 Areas of Regular Polygons

33. 11.3 Areas of Regular Polygons
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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11.3 Areas of Regular Polygons

34. 11.3 Areas of Regular Polygons
Homework
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11.3 Areas of Regular Polygons