Circles in Experiments

​Geometry Richness Always Demands Exploring

CIRCLES IN EXPERIMENTS eTwinning Project 2019-2020

Coord.Teachers: Mihaela Gîț, Katarina Ivanovic, Ludmila Cojocari „ Jean Monnet” High School-Bucharest, Primary School Stevan Dukic-Belgrade, Gaudeamus High School-Chișinău

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Task Problem

1. Easter Egg Construct Easter egg, calculate its Circumference and Area.

The Team who draw Drawing/solution

The Team who solve Solution

Serbia

Romania Team 1

2. Japanese temple During the Edo period (1603-1867) of Japanese history, geometrical puzzles were hung in the holy temples as offerings to the gods and as challenges to worshippers. This is one such problem. Inside a large circle with radius 2r, two circles of radius r are drawn. Four smaller circles, of radius p, are drawn to touch the large circle and the circles of radius r.

Romania 2

Romania Team 2

OK= x = EF EO =2r – x; EC= r + x CK = r – x ΔEKC : E K 2 = E C 2 –CK 2 ΔEKC : E K 2 = E C 2 –CK 2 ΔEKO : E K 2 = E O2 –OK 2 (r+x)​2​ – (r–x)​2 ​=(2r–x)​2 ​–x​2 (r+x+r–x)(r+x–r+x) =(2r–x+x)(2r–x–x) 2r · 2x =2r (2r–2x) ⇒ x = r–x ⇒ 2x=r ⇒ x= 2r

Romania Team 3

3. Love is in the air a) What is the area percentage of square in a heart? b) What is that ratio of circumference of square and circumference of the heart?

Let R the radius of the circles and L the side of the square.

L=2R The area of the square is L2 = 4R2 The area of the heart is: 2πR2 + 4R2 − π R2 = π R2 + 4R2 A square 4R2 4 = 4+π 2 2 ⇒ A heart πR +4R

The circumference of the heart is: 3 1 4 · 2πR · 2 + 4 · 2πR · 2 = 4πR The circumference of the square is: L=2R ⇒ P = 4L = 8πR. 4πR 8πR

=

1 2

Serbia

Romania Team 4

4. Shaded area ​What

fraction of semicircle is shaded​?

R.Moldova Team 1

5. Stained Glass Design You need to make this stained-glass window to hang in the school entryway. The window will be 2 m tall and 5 m wide.

We would give you 100 euros for the materials for the project. The colored glass costs 5 euros per square meter and the clear glass costs 3 euros per square meter.

The materials you need to join the pieces of glass together costs 10 cents per meter and the frame costs 4 euros per meter.

Do you have enough money to cover the costs of the materials you will need to make the window?

Area rectangle = 10 m​2 Diameter circle is 1.

Radius of a circle is 1/2 14 white semicircles+12white quarter circles = =10 white circles White area=10 π · 14 =7.85 Colored –7.85=2.15

Area=

10–10 π · 14 =10

Serbia Team 1

6.

Pizza coupons If you'd like the most pizza, which coupon should you use?

So, would the best coupon for the slice above work for all slices or just some slices?

Under what circumstances you should use one coupon or the other?

R.Moldova Team 2

Serbia Team 1

7. Circle Packing What percentage of the plane is covered by circles in each packing pattern?

How is this of use in packing cylindrical cans and what are the advantages and disadvantages of the two packing systems?

Serbia Team 2

Romania Team 5

8. Fidget spinner

Romania Team 6

R. Moldova Team 3

Each of the following arclets is made from arcs of a circle of radius r.

What is the perimeter of an arclet with 3, 4, 5, and n "nodes"?

P 3 = 3 · 2πR = 6πR

Serbia Team 3 P 3 = 3 · 2πR = 6πR P 4 = 4 · 2πR = 8πR P 5 = 5 · 2πR = 10πR P n = n · 2πR = 2πnR

9. B&W world

R.Moldova Team 3& Romania Team 4

Serbia Team 1

Calculate the Circumference and Area of the BLACK (shaded) parts on image, if the area of the square is 100.

If L is the side of the square ABCD ⇒ L2 = 100 ⇒ L = 10 .

L=2R (the radius of the great circle) ⇒R=5 A​Black​=​A​White = A ⇒ A = ⇒A=

25π 2

πR2 2

=

25π 2

.

R1 = 1; R2 = 2; R3 = 3; R4 = 4; 2πR = π R = π · 5 = 5π ; 2 Arcs circumference: π R1 = π ; πR2 = 2π; π R3 = 3π; π R4 = 4 All Circumference: C= π +2 π +3 π +4 π +5 π +8 · 5 = 15π + 40 ⇒ C=15 π +40

10.

Yin and Yang

Romania Team 1

Romania Team 2

... siymbol also known as a Taijitu symbol, with two parts: white representing Yang and black representing Yin. Calculate Circumference and Area of shaded part.

2R=a= 8x ; R=4x; r=2x; r’=​x ⇒ C (D, x) = C (E, x) and C (D, 2x) = C (E, 2x); C(D, 4x) = C (E, 4x) 2 White Area = Shaded Area= π R2 = πa4

Shaded area=

πa2 4

Red circumference= 2πr + 2 · 2πr′ = 2π · a4 + 2π ·

Red circumference =

a 8

3aπ 4

= 2π ·

3a 8

=

3aπ 4

11.

Yin and Yang Diameter AE of the circle is divided at C in the ratio 2 : 3.

The two semicircles ArcABC and ArcCDE divide the circular region into an upper (shaded) region and a lower region. Find the ratio of the area of the upper region to that of the lower.

R. Moldova Team 4

SerbiaTeam 1