Try this beautiful problem from Geometry based on Pentagon and square Pattern.

Pentagon & Square Pattern – AMC-10A, 2001- Problem 18


The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to

pentagon and square pattern
  • \(50\)
  • \(58\)
  • \(60\)
  • \(56\)
  • \(64\)

Key Concepts


Geometry

Pentagon

square

Check the Answer


But try the problem first…

Answer: \(56\)

Source
Suggested Reading

AMC-10A (2001) Problem 18

Pre College Mathematics

Try with Hints


First hint

pentagon and square pattern

The given square is the above square.we have to find out The percent of the plane that is enclosed by the pentagons.Notice that there are \(9\) tiles in the square box.so if we can find out the area of pentagon and small square in single tile,then we can find out the total area of the pentagon in the total big square……

can you finish the problem……..

Second Hint

Shaded pattern

Now consider a single tile from the big square,Let us take the side of the small square is $a$.There are four squares which is in the area \(4a^2\) and there are five pentagons which are in areas \(5a^2\).Then the area of the single tile is $9a^2$

Therefore we can say that exactly $5/9$ of any tile are covered by pentagons, and therefore pentagons cover $5/9$ of the plane also

can you finish the problem……..

Final Step

pattern

Therefore for the whole square, expressed as a percentage,it becomes $55.\overline{5}\%$, and the closest integer to this value is \(56\)

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