1193. To fly!

    Teacher, thanks for teaching me how to fly.

    I gave him back the compliment... I tell Pepe Chapuzas that it was very easy to teach to fly to children who were born with wings...
    Pepe said goodbye with a gift: a Geometry problem...

    Dear Teacher:
    The orange area is 10 m², and the green area is...

    Nina Guindilla was born with wings too...

  

    Dear Teacher:

    There are 10 orange circles, so, one circle area is  1 m² , and its radius measures  1/√π .

    The radius of the blue decagon measures  csc 18° / √π . I have no calculator but I have here a chart of cosecants...

csc 15°  =  √6 + √2

csc 18°  =  √5 + 1

csc 30°  =  2

csc 45°  =  √2

csc 54°  =  √5 − 1

csc 60°  =  √12 / 3

csc 75°  =  √6 − √2

csc 90°  =  1

    So, the radius of the blue decagon measures

(√5 + 1) / √π .

    And the radius of the green circle measures

(√5 + 1) / √π  −  1 / √π  =  √5 / √π .

    And the green area is... half the orange area

π · (√5 / √π)²  =  5 m² .