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Train track shapes (Posted on 2011-08-01) Difficulty: 3 of 5
I have a toy train set with lots of curved pieces. If you put any 8 of these pieces together you can make a circle with radius r.

Ignoring the width of the track, find the area (in terms of r) enclosed by each of the following configurations:

1) As the train goes around the track clockwise it goes 5 turns right then 1 turn left. This pattern continues twice.

2) 6 right 2 left, twice.

3) 5 right 1 left 1 right 1 left, twice.

4) 3 right 1 left, four times.

5) 4 right 1 left 2 right 1 left, twice.

6) 5 right 1 left 2 right 1 left 5 right 2 left.

Bonus: Are there any other configurations using 16 or fewer of these pieces?

See The Solution Submitted by Jer    
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an answer to 1 thru 6 Comment 2 of 2 |

Each of the eight sections of track subtends a sector of a circle formed by the track of an area equal to pi*R2/8. Henceforth, the sector formed will be referred to as a "slice".

1. The area of the curved geometric planar shape is equal to the area of 5 "slices", plus a rhombus with base of 2*R and height of sqrt(2)*R minus 2 "slices", plus 5 "slices"; which is equal to
(pi + 2*sqrt(2))*R2.

2. The area of the curved geometric planar shape is equal to the area of 6 "slices", plus a "tilted" square with both base and height of 2*R minus 4 "slices", plus 6 "slices";  which is equal to (pi + 4)*R2.

3. The area of the curved geometric planar shape is equal to the area of 5 "slices", plus a rhombus with base 2R and height of sqrt(2)R minus 2 "slices", plus 2 "slices", plus another rhombus with base 2*R and height of sqrt(2)*R minus 2 "slices", plus 5 "slices"; which is equal to
(pi + 4*sqrt(2))*R2.

4. The area of the curved geometric planar shape is equal to the area of 3 "slices", plus a half of a rhombus with base of 2R and height of sqrt(2)R minus 2 "slices" -- four times, plus the area of a square with a side of 2*sqrt(2 - sqrt(2))*R; which is equal to (pi + 8)*R2.

5. The area of the curved geometric planar shape is equal to the area of 4 "slices" plus two halves of a rhombus with base of 2R and height of sqrt(2)R minus 2 "slices", plus 2 "slices" -- two times; plus a rhombus with side of 2*sqrt(2 - sqrt(2))*R and sine of the supplementary angles equal to sqrt(2)/2; which is equal to
(pi + 8*sqrt(2) - 4)*R2.

6. The area of the curved geometric planar shape is equal to the area of 6 "slices", plus two rhombi with base 2R and height sqrt(2)R with each minus 2 "slices", plus 6 "slices"; which is equal to (pi + 4*sqrt(2))*R2.

Bonus:
Besides the 6 configurations listed, there is the 1 trivial configuration mentioned in the first paragraph -- that of the circle of area pi R2.

Edited on August 5, 2011, 1:46 am

Edited on August 5, 2011, 2:16 am
  Posted by Dej Mar on 2011-08-03 03:08:48

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