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RIP-ROAR TRAINING & CONSULTING By Leland Bartlett

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Rip-Roar Training & ConsultingByLeland Bartlett

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Todays Proof is how:

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How we are going to proceedReminder of Area FormulasHow the idea came to meIts all in the trianglesQuick Trigonometry LessonParameters to work towardsArea of 4 triangles8 triangles16 triangles32 triangles1024

SummaryAgreementProofTa-dahAdditional InfoMy Info

Remember these formulas?SquareRectangleS1S2S1S2

Trianglehb

CircleA=S1 * S2A=S1 * S2A = * r2

4

octagonIts all in the angles, the tri-angles

dodecagon

square

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Its all in the angles, the tri-angles

12345678

123456789101112

1234

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Triangles and some Trigonometry In order to calculate the triangles, a couple trigonometry functions. Sin( ) and Cos( )

ASin(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 . 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)eighthypotenusesBaseaIn 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

45o22.o90o

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What we are shooting for. . .

D = 6

r = 3Diameter = 6 inchesRadius = 3 inches

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4 -- TrianglesBase:Sin(45) = b/3 Base = 4.242640687 Height:COS(45) = h/3 h = 2.121320344 Area:4*[ (4.242640687) * (2.121320344)] = 18

Difference: 10.27Circumference:4* 4.242640687 = 16.97056275

Difference = 1.878993173

900

450b

3

Baseh

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8 -- TrianglesBase:Sin(22.5) = b/3 Base = 2.296100594 Height:Cos(22.5) = h/3 h = 2.771638598 Area:8*[ (2.296100594 ) * (2.771638598 )] = 25.45584

Difference: 2.81848976 Circumference:8* 2.296100594 = 18.36880475

Difference: 0.480751168

2456781

45

22.5

Base

b

3h

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16 -- TrianglesBase:Sin(11.25) = b/3 Base = 1.170541932 Height:Cos(11.25) = h/3 h = 2.942355841 Area:16*[ (1.170541932 ) * (2.942355841 )] = 27.55321

Difference: 0.721126752 Circumference:16* 1.170541932 = 18.72867091

Difference: 0.120885008

23456781910111213141516

22.5

11.25

Base

b

3h

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32-- TrianglesBase:Sin(5.625) = b/3 Base = 0.588102842 Height:Cos(5.625) = h/3 h = 2.98555418 Area:32*[ (0.588102842) * (2.98555418)] = 28.09301

Difference: 0.181327512 Circumference:32* 0.588102842 = 18.81929094

Difference: 0.030264978

2345678191011121314151617181920212223242526272829303132

11.25

5.625

Base

b

3h

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1024- TrianglesBase:Sin(0.17578125) = b/3 b = 0.018407741 Height:Cos(0.17578125) = h/3 h = 2.999985881 Area:1024*[ (0.018407741) * (2.999985881)] = 28.27416

Difference: 0.0001774190043129Circumference:1024* 0.018407741 = 18.84952635

Difference: 0.00002956987579949550

2345678191011121314151617181920212223242526272829303132

32 Triangles in each section

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Table of Values

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If we could sum up all the triangles. . .

D = 6

r = 3Radius = 3Height = 2.999985881Diff of = .000014119

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Would you agree. . .

D = 6

r = 3Radius = 3Height = 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?

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So, if you agree This is How it works? 2r h = r2C = D * D = 2r b h = r2 C h = r2 D h = r2

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How does it work final Steps? r2 = r2h = 2.999985881h r

D = 6

r = 3 r h = r2 r r = r2 r2 = r2

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Ta-dah!!! r2 = r2

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Additional Info . . .

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Leland Bartlett SummaryI 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:instructor-led-trainingindividual training eLearningOn-the-job trainingBlended learningSelf-paced learning

Microsoft OfficeExcel AdvancedAccess AdvancedWord AdvancedPowerPoint AdvancedOneNote AdvancedOutlook AdvancedVisionProjectOthersMSSQLAdobe CaptivateCamtasia . . .more

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Questions?Contact Leland Bartlett by:Phone: 682-667-1243 email: [email protected] me a comment in the comment section of this video.

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