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 :

=  s

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

1/2 ⋅ d2  =  32

Multiply each side 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 :

=  s

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

Perimeter of 1st square

=  4(4)

=  16 cm

Perimeter of 2nd square

=  4(5)

=  20 cm

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

Area of 1st square

=  42

=  16 cm2

Area of 2nd square

=  52

=  25 cm2

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 

=  √[(x- x1)+ (y- 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)+ (9 - 2)2]

=  √[6+ 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 

=  √[(x- x1)+ (y- 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)+ (8 - 4)2]

=  √[3+ 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 :

=  1/2 ⋅ d2

Substitute d  =  5. 

=  1/2 ⋅ 52

Simplify. 

=  1/2 ⋅ 25

=  12.5

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

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