PERIMETER AND AREA OF SQUARE WORKSHEET

Problem 1 :

If the length of each side of a square is 8.5 cm, then find its perimeter.

Problem 2 :

The length of each diagonal of a square is 2√2 cm. Find its perimeter.

Problem 3 :

If a square has the side length of 7.5 cm, then find its area.

Problem 4 :

The length of each side of a square is 3√5 cm. Find its area.

Problem 5 :

The area of a square is 32 square inches. Find the length of its diagonal.

Problem 6 :

The square has side length 36 inches. Find its area in square feet.

Problem 7 :

The lengths of each side of two squares are 4 cm and 5 cm. Find the ratio of their perimeters.

Problem 8 :

The lengths of each side of two squares are 4 cm and 5 cm. Find the ratio of their areas.

Problem 9 :

AB is one of the sides of the square ABCD and the side AB is defined by A(0, 2) and B(6, 9). Find the perimeter of the square ABCD.

Problem 10 :

PR is one of the diagonals of the square PQRS and the diagonal PQ is defined by P(1, 4) and Q(4, 8). Find the area of the square PQRS.

Answers

1. Answer :

Formula for perimeter of a square :

= 4s

Substitute 14 for s.

= 4(8.5)

= 34

So, the perimeter of the square is 34 cm.

2. Answer :

To find the perimeter of a square, first we have to know the length of each side.

Let s be the length of each side of the square.

Draw a sketch.

In the figure shown below, consider the right triangle ABC.

By Pythagorean Theorem, we have

AB² + BC² = AC²

Substitute.

s² + s² = (2√2)²

Simplify and solve for s. 

2s² = 2² ⋅(√2)²

2s² = 4 ⋅(2)

2s² = 8

Divide each side by 2. 

s² = 4

Find positive square root on both sides.

 √s² = √4

 √s² = √(2 ⋅ 2)

s = 2

Formula for perimeter of a square. 

Perimeter = 4s

Substitute 2 for s.

= 4(2)

= 8

So, the perimeter of the the square is 8 cm.

3. Answer :

When the length of a side is given, formula for area of a square :

= s² 

Substitute 24 for s.

= 7.5²

= 56.25

So, area of the square is 56.25 square cm.

4. Answer :

When the length of a side is given, formula for area of a square :

= s²

Substitute 3√5 for s.

= (3√5)²

Simplify.

= 3² ⋅ (√5)²

= 9 ⋅ 5

= 45

So, the area of the square is 45 square cm.

5. Answer :

Area of the square = 32 in²

½ ⋅ d² = 32

Multiply both sides by 2.

d² = 64

Find positive square root on both sides.

 √d² = √(8 ⋅ 8)

  d = 8

So, the length of diagonal is 8 inches. 

6. Answer :

When the length of a side is given, formula for area of a square :

= s² 

Substitute 12 for s.

= 36²

= 1296 in² -----(1)

We know  

12 inches = 1 ft

Square both sides.

(12 inches)² = (1 ft)²

12² in² = 1² ft²

144 in² = 1 ft²

Therefore, to convert square inches into meter square feet,  we have to divide by 144. 

(1)-----> Area of the square = 1296 in²

Divide the right side by 144 to convert in² into ft².

Area of the square = 1296/144 ft²

= 9 ft²

So, the area of the square is 9 square feet. 

7. Answer :

Formula for perimeter of a square :

= 4s

Ratio of the perimeters :

= 16 : 20

Divide each term by 4.

= 4 : 5

So, the ratio of the perimeters of two squares is 4 : 5.

8. Answer :

Formula for area of a square :

= s²

Ratio of the areas :

= 16 : 25

So, the ratio of the areas of two squares is 16 : 25.

9. Answer :

Distance between the two points (x₁, y₁) and (x₂, y₂) is 

= √[(x₂ - x₁)² + (y₂ - y₁)²]

To find the distance between A and B, substitute 

(x₁, y₁) = (0, 2)

(x₂, y₂) = (6, 9)

in the above formula.

Distance between A and B : 

= √[(6 - 0)² + (9 - 2)²]

= √[6² + 7²]

= √[36 + 49]

= √85

Therefore, the length of one of the sides is √85 units.

Formula for perimeter of a square :

= 4s

Substitute s = √85. 

= 4√85

Use calculator and simplify.

≈ 36.88

So, the perimeter of the square ABCD is about 36.88 units.

10. Answer :

Distance between the two points (x₁, y₁) and (x₂, y₂) is 

= √[(x₂ - x₁)² + (y₂ - y₁)²]

To find the distance between P and R, substitute 

(x₁, y₁) = (1, 4)

(x₂, y₂) = (4, 8)

in the above formula.

Distance between P and R : 

= √[(4 - 1)² + (8 - 4)²]

= √[3² + 4²]

= √[9 + 16]

= √25

= 5

Therefore, the length of the diagonal PR is 5 units.

When the length of a diagonal is given, formula for area of a square :

= ½ ⋅ d²

Substitute d  =  5. 

= ½ ⋅ 5²

Simplify. 

= ½ ⋅ 25

= 12.5

So, the area of the square PQRS is 12.5 square units.  

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