# 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. 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. 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.