Geometry Problems on Squares

Word problems on squares with detailed solutions.

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.

    We now present word problems with detailed 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

    More references on geometry.


    Geometry Tutorials, Problems and Interactive Applets.

    Navigation

    Search