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Geometry of mathematics
http://www.sciencebuddies.org/science-fair-
projects/project_ideas/Math_p010.shtml
s1230246 Keita wagatsuma
What is a geometry ?
• geometry is a field of mathematics.
• geometry treats graphics mainly.
Math formula and geometry(1)
・Math formula and geometry have   
 important relationship.
・Math formula is a simple way to  
 represents geometry.
Math formula and geometry(2)
• For example, we replace x=r*cosθ,y=r*sinθ
in circle. so,we can express geometry using
math formula.
Math formula and geometry(3)
・x^2 plus y^2 equals r^2.
Because (sinθ)^2+(cosθ)^2 = 1
・We can prove this problem
solving formula.
x^2+y^2 = (r*cosθ)^2+(r*sinθ)^2
= r^2(sinθ^2+cosθ^2)
= r^2
i.e. x^2+y^2 = r^2 Q.E.D
what is a circle ?
• Circle have diameter and radius and
center and Circumference.
• Circle have important official.
Important circle official
• Circle have important official that you
solve the shape of problem.
Circumference = 2 × π × Radius
= 2 × π × Diameter
How to prove circle official (1)
Circumference = 2 × π × Radius
・This is a famous circle official.      
but,everyone doesn’t know how to prove it.
・We can prove it using an extreme way of
thinking.
How to prove circle official (2)
• We divide the circle into n triangles.
• We should think about one triangle.
• And, we create math formula.
How to prove circle official(3)
We define things that radius is r and the
bottom of one of the triangle is l and
circumference is L.
l = 2*r*sin(π/n)
L = n*l = 2*π*r*(sin(π/n)/(π/n))
i.e. L = 2*π*r (n→∞) Q.E.D
What is the Pi ?
• Pi is special number.
• It is Irrational number.
• It has been used in various fields of analysis
and mechanics.
• This figure represents main problem.
• This site’s main problem is thing that We
have to prove AE = AB + BC + CD + DE
Main problem that this site
• We define things that AB is l1 and AB plus
BC plus CD plus DE is l2.
• Diameter of AE is R.
• Radius of AB is r1. And Similarly, I define it.
How to solve this problem(1)
How to solve this problem(2)
We can find simple relationship.
R = 2*(r1+r2+r3+r4)…①
Next,l1 = (2*π*(1/2)*R)*(1/2)
= (1/2)*π*R
14
How to solve this problem(3)
We can calculate l2 using basic circle official.
l2 =
(2*π*r1)*(1/2)+(2*π*r2)*(1/2)+(2*π*r3)*(1/2)+(2*π
*r4)*(1/2)
=π*r1+π*r2+π*r3+π*r4
=π(r1+r2+r3+r4)
=π*(1/2)*R (Use ①)
i.e. l1=l2
∴ AE = AB + BC + CD + DE Q.E.D
15

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Englush assignment c_s1230246_keitawagatsuma

  • 2. What is a geometry ? • geometry is a field of mathematics. • geometry treats graphics mainly.
  • 3. Math formula and geometry(1) ・Math formula and geometry have     important relationship. ・Math formula is a simple way to    represents geometry.
  • 4. Math formula and geometry(2) • For example, we replace x=r*cosθ,y=r*sinθ in circle. so,we can express geometry using math formula.
  • 5. Math formula and geometry(3) ・x^2 plus y^2 equals r^2. Because (sinθ)^2+(cosθ)^2 = 1 ・We can prove this problem solving formula. x^2+y^2 = (r*cosθ)^2+(r*sinθ)^2 = r^2(sinθ^2+cosθ^2) = r^2 i.e. x^2+y^2 = r^2 Q.E.D
  • 6. what is a circle ? • Circle have diameter and radius and center and Circumference. • Circle have important official.
  • 7. Important circle official • Circle have important official that you solve the shape of problem. Circumference = 2 × π × Radius = 2 × π × Diameter
  • 8. How to prove circle official (1) Circumference = 2 × π × Radius ・This is a famous circle official.       but,everyone doesn’t know how to prove it. ・We can prove it using an extreme way of thinking.
  • 9. How to prove circle official (2) • We divide the circle into n triangles. • We should think about one triangle. • And, we create math formula.
  • 10. How to prove circle official(3) We define things that radius is r and the bottom of one of the triangle is l and circumference is L. l = 2*r*sin(π/n) L = n*l = 2*π*r*(sin(π/n)/(π/n)) i.e. L = 2*π*r (n→∞) Q.E.D
  • 11. What is the Pi ? • Pi is special number. • It is Irrational number. • It has been used in various fields of analysis and mechanics.
  • 12. • This figure represents main problem. • This site’s main problem is thing that We have to prove AE = AB + BC + CD + DE Main problem that this site
  • 13. • We define things that AB is l1 and AB plus BC plus CD plus DE is l2. • Diameter of AE is R. • Radius of AB is r1. And Similarly, I define it. How to solve this problem(1)
  • 14. How to solve this problem(2) We can find simple relationship. R = 2*(r1+r2+r3+r4)…① Next,l1 = (2*π*(1/2)*R)*(1/2) = (1/2)*π*R 14
  • 15. How to solve this problem(3) We can calculate l2 using basic circle official. l2 = (2*π*r1)*(1/2)+(2*π*r2)*(1/2)+(2*π*r3)*(1/2)+(2*π *r4)*(1/2) =π*r1+π*r2+π*r3+π*r4 =π(r1+r2+r3+r4) =π*(1/2)*R (Use ①) i.e. l1=l2 ∴ AE = AB + BC + CD + DE Q.E.D 15