3. HOW WE ARE GOING TO PROCEED
• Reminder of Area Formulas
• How the idea came to me
• It’s all in the triangles
• Quick Trigonometry Lesson
• Parameters to work towards
• Area of
• 4 triangles
• 8 triangles
• 16 triangles
• 32 triangles
• 1024
• Summary
• Agreement
Proof
• Ta-dah
• Additional Info
• My Info
6. IT’S ALL IN THE ANGLES, THE TRI-ANGLES
1
2
3
4
5
6
7
8
1 2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
7. TRIANGLES AND SOME TRIGONOMETRY
In order to calculate the triangles, a couple trigonometry functions. Sin( ) and Cos( )
A
Sin(A) = the side opposite of the angle over the hypotenuses. The hypotenuses is the longest side of a
triangle. So we have to divide the triangles in ½. This is also needed to be done in order to calculate the
height of the triangle. Additionally, when doing the calculation for the area of the triangle, the base needs to
be multiplied by 2 in order to get the area of the triangle correct. The cos(A) would use the adjacent side, the
height over the hypotenuses. Also, note, the angle A needs to be divided in half as well. So, an angle of 90
deg, the angle used with sin and cos would be 45 deg.
(h)eight
Base
a
In the triangle There is angle A the whole angle then the angle a being used to
calculate the height and base. In the example, A = 45 deg and then angle a =
22.5 deg. The sides of the angle are equal to the radius from our circle or 3.
Sin(a) = Opp/Hyp Sin(22.5) = b/3 b = 3(sin(22.5))
Cos(a) = Adj/Hyp Cos(22.5) = h/3 h = 3(cos(22.5))
b
3
45o
22.o
90o
8. WHAT WE ARE SHOOTING FOR. . .
D = 6
r = 3
Circumference = 𝜋* 6 => 18.8495559215 sq in
Area = 𝜋 * r2 => 𝜋 * 9 => 28.274333882308 sq in
Diameter = 6 inches
Radius = 3 inches
15. IF WE COULD SUM UP ALL THE TRIANGLES. . .
D = 6
r = 3
Circumference = 𝜋* 6 => 18.8495559215
Calculated 1024 = 18.84955589
Area = 𝜋 * r2 => 𝜋 * 9 => 28.274333882308
Calculated: 28.27415646
Radius = 3
Height = 2.999985881
Diff of = .000014119
16. WOULD YOU AGREE. . .
D = 6
r = 3
Circumference = 𝜋* 6 => 18.8495559215
Sum of Bases<1024> = 18.84955589
Area = 𝜋 * r2 => 𝜋 * 9 => 28.274333882308
Sum of the area of Triangles: 28.27415646
Radius = 3
Height = 2.999985881
• The sum of the base of the triangles is a good approximation
of the circumference of the circle?
• The height of the triangle is a pretty good approximation of
the radius of the circle?
17. SO, IF YOU AGREE – THIS IS HOW IT WORKS?
½ 2r h = r2
Circumference = 𝜋* 6 => 18.8495559215
Sum of B <1024> = 18.84955589
C = D *
D = 2r
½ b h = r2
½ C h = r2
½ D h = r2
18. HOW DOES IT WORK – FINAL STEPS?
r2 = r2
h = 2.999985881
h ≈ r
D = 6
r = 3
2𝑟 h
2
= r2
r h = r2
r r = r2
21. LELAND BARTLETT SUMMARY
• I am a professional consultant, consultant trainer and trainer. I've been doing both
soft skills and hard skills training for over 30 years.
• I provide the following training:
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22. QUESTIONS?
• Contact Leland Bartlett by:
• Phone: 682-667-1243
• email: leland.bartlett@outlook.com
• Leave me a comment in the comment section of this video.