Question

Eight coins fit exactly on a rectangular sheet of paper when lined up in 4 columns and two rows. Each coin is in the shape of a regular dodecagon, a polygon containing 12 sides. What fraction of the paper sheet will not be covered?

Answer 1

`A_(f)`≅0.034
`L^(2)`

Explanation:

The graph shows a rectangle and 8 dodecahedrons but a dodecahedron inscribed in a square is seen, so it is enough to calculate the fractions of areas between the square and the dodecahedron

Square area
`A_(s)=0.134L^(2)`

Area of ​​a right triangle
`A_(rt)=((0.232L*0.134L))/(2)`

Area fraction
`A_(f)=4A_(c)+8A_(rt)\\A_(f)=4*(0.134L)^(2)+8*(0.232L*0.134L)/(2) =0.017956L^(2)+0.015544L^(2)`

`A_(f)`≅0.034
`L^(2)`

Note: The fractions of the sides of the square and the right triangle are determined by assigning values ​​to L