Geometry Problems on Squares

Problems on area and perimeter of squares with detailed solutions are presented.

Area and Perimeter of a Square

Problems with Deatiled Solutions

Problem 1

• Let S be the side length before the increase, the area A1 is given by
  A1 = S ²
  
• Let S + 2 the side after the increase, the area A2 is given by
  A2 = (S + 2) ²
  
• But A2 = A1 + 44, hence
  A1 + 44 = (S + 2) ²
  
• Substitute A1 by S ² in the above equation.
  S ² + 44 = (S + 2) ²
  
• Expand, group like terms and rewrite the equation as follows.
  4 S = 40
  
• Solve for S.
  S = 10 feet.
  
• As an exercise, find areas for S = 10 and for S = 12 and check the the difference is 44 feet ².

Problem 2

• Use Pythagora's theorem to write
  S ² + S ² = 200 ²
  
• Solve for S ² to find the area
  S ² = 20000 m ²
  
• We need to find S to find the perimeter
  S = 100 sqrt(2)
  Perimeter = 4 S = 400 sqrt(2) m.

Problem 3

• The area A1 of a square of side length S is given by.
  A1 = S ²
  
• Double the side to 2S and find the new area A2.
  A2 = (2 S) ² = 4 S ²
  
• The area is multiplied by 4.

Problem 4

• The total area A of the garden and the walkway is given by.
  A = 400 + 500 = 900 m ²
  
• The side S of the garden is given by.
  S = sqrt(400) = 20 m
  
• The outside of the walkway is a square of side
  S + x + x = S + 2x = 20 + 2x.
  
• The total area of the large square is equal to 900 m ², hence the equation:
  (20 + 2x) ² = 900
  
• We now solve for x
  x = 5 and x = -25
  
• x is a measure of length and has to be positive, hence
  x = 5 meters.
  
• As an exercise, find the side of the larger square and its area and check with the total value of the area 900 m. ²

More References and Links to Geometry Problems