This is a script that geometry of mathematics.

1. Geometry of mathematics
http://www.sciencebuddies.org/science-fair-
projects/project_ideas/Math_p010.shtml
s1230246 Keita wagatsuma

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
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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
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