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

- \(50\)
- \(58\)
- \(60\)
- \(56\)
- \(64\)

**Key Concepts**

Geometry

Pentagon

square

## Check the Answer

Answer: \(56\)

AMC-10A (2001) Problem 18

Pre College Mathematics

## Try with Hints

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……..

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……..

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

## Other useful links

- https://www.cheenta.com/medians-of-triangle-prmo-2018-problem-10/
- https://www.youtube.com/watch?v=fRj9NuPGrLU&t=265s