The corners of a square with side 10 is cut away to form an octagon with all sides equal. What is the length of a side of the octagon so formed?


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Answer:

 

10
1 + √2
 

Step by Step Explanation:
  1. Let ABCD be a square and PQRSTUVW is a octagon with equal sides constructed within a square.
  2. As we have to construct an octagon with equal sides we will take AP = AQ = QB = BR = BS = UC = CT = DV = DW.
  3. Let AP = AQ = x [Side of the square cut]
    and PQ = y [Side of the regular octagon]
  4. As all sides of the octagon are equal.
    QR = RS = ST = TU = UV = VW = WP = PQ = y
  5. A B C D P W Q R S T U V x y x
  6. As the length of the square is 10 cm,
    2x + y = 10 ................(1)
  7. Applying Pythagoras to the right angled triangle APQ,
    y2 = x2 + x2 = 2x2
    or y = x √2...............(2)
  8. Solving (1) and (2), we get y =  
    10
    1 + √2
     .
  9. Thus, the length of the side of the octagon so formed is  
    10
    1 + √2
         .

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