A square is a four-sided closed figure where the lengths of all the four sides will be equal and each vertex angle will be right angle or 90^{o }as shown below.

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

Then, the formula for area of a square :

**Area = s ^{2}**

Let d be the length of each diagonal of a square.

Then, the formula for area of a square :

**Area = 1/2 ****⋅ d ^{2}**

**Example 1:**

Find the area of the square having side length 24 cm.

**Solution:**

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

= s^{2 }

Substitute 24 for s.

= 24^{2}

= 576

So, area of the square is 576 square cm.

**Example 2:**

If the area of a square is 64 square inches, then find the length of each side.

**Solution:**

Area of the square = 64 in^{2}

S^{2} = 64

Find positive square root on both sides.

√s^{2} = √(8 ⋅ 8)

S = 8

So, the length of each side of the square is 8 inches.

**Example 3:**

The square has side length 250 cm. Find its area in square meter.

**Solution:**

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

= s^{2 }

Substitute 250 for s.

= 250^{2}

= 62500 cm^{2} -----(1)

We know

100 cm = 1 m

Square both sides.

(100 cm)^{2} = (1 m)^{2}

100^{2} cm^{2} = 1^{2} m^{2}

10000 cm^{2} = 1 m^{2}

Therefore, to convert centimeter square into meter square, we have to divide by 10000.

(1)-----> Area of the square = 62500 cm^{2}

Divide the right side by 10000 to convert cm^{2} into m^{2}.

Area of the square = (62500 / 10000) m^{2}

= 6.25 m^{2}

So, the area of the square is 6.25 square meter.

**Example 4:**

If the length of each diagonal is 2√2 cm, then find its area.

**Solution:**

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

= 1/2 ⋅ d^{2}

Substitute 2√2 for d.

= 1/2 ⋅ (2√2)^{2}

Simplify.

= 1/2 ⋅ (4 ⋅ 2)

= 1/2 ⋅ (8)

= 4

So, the area of the square is 4 square cm.

**Example 5:**

If the lengths of the diagonals of two squares are in the ratio 2 : 5. then find the ratio of their areas.

**Solution:**

From the ratio 2 : 5, let the diagonals of two squares be 2x and 5x respectively.

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

= 1/2 ⋅ d^{2}

= 1/2 ⋅ (2x) = 1/2 ⋅ (4x = 4x |
= 1/2 ⋅ (5x) = 1/2 ⋅ (25x = 25x |

Ratio of the areas :

= (4x^{2} / 2) : (25x^{2} / 2)

Multiply each term of the ratio by 2.

= 4x^{2} : 25x^{2}

Divide each term by x^{2}.

= 4 : 25

So, the ratio of the areas of two squares is 4 : 25.

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