Geometry Problems on Squares

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

Area and Perimeter of a Square

Perimeter of a Square
Perimeter = 4 S , S is the side length of the square.
Area of a Square
Area = S 2 , S is the side length of the square.

Problems with Deatiled Solutions

Problem 1

When the sides of a square are each increased by 2 feet its area increases by 44 feet 2. Find the side length S before the increase.

Solution to 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

Find the area and perimeter of a square with diagonal of 200 meters.
square problem 2

Solution to 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

What happens to the area of a square if we double its side?

Solution to 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

A square garden (green) of 400 m 2 is to be surrouned by a walkway (yellow) of constant width x. The total area of the walkway has to be 500 m 2. Find the width x of the walkway.
square problem 4

Solution to 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

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