What is the formula for the area of a square?

The area of a square is calculated by multiplying the side length by itself: Area = side x side, or Area = s². For example, a square with a side length of 5 cm has an area of 25 cm². The unit of the result is always the square of the unit used for the side (so cm gives cm², m gives m², and so on).

How do you find the perimeter of a square?

The perimeter of a square is the total distance around all four sides: Perimeter = 4 x side length. For a square with side 5 cm, the perimeter is 4 x 5 = 20 cm. Because all four sides of a square are equal, you simply multiply the side length by 4.

How do you calculate the diagonal of a square?

The diagonal of a square is found using the Pythagorean theorem. Because a square has four equal sides and four right angles, the diagonal d = s x sqrt(2), where s is the side length. For a square with side 5 cm, the diagonal is 5 x 1.41421 = approximately 7.071 cm. The diagonal splits the square into two right-angled triangles where the diagonal is the hypotenuse.

Area of a Square Calculator

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Standard formula Area = s², Perimeter = 4s, Diagonal = s√2. Standard Euclidean geometry.

1. Square Dimensions

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2. Worked Example

For a square with side length 5 m:

Area = s²5 × 5 = 25 m²

Perimeter = 4s4 × 5 = 20 m

Diagonal = s√25 × 1.41421 = 7.071 m

The worked example updates with your inputs.

Results

Area
25
m²

Perimeter

Diagonal

7.071

Full Breakdown

Side length (s)5 m

s² (area)25 m²

4 × s (perimeter)20 m

s × √2 (diagonal)7.0711 m

Diagonal / side ratio√2 ≈ 1.41421

Formulas Used

AreaA = s²

PerimeterP = 4s

Diagonald = s√2

Side from areas = √A

Side from perimeters = P ÷ 4

Side from diagonals = d ÷ √2

Summary: A square with side 5 m has an area of 25 m², a perimeter of 20 m and a diagonal of approximately 7.071 m.

How to Calculate the Area of a Square

A square is a regular quadrilateral: it has four equal sides and four right angles (each 90 degrees). Because all four sides are the same length, only one measurement is needed to fully describe a square.

The three key measurements of a square all follow from the single side length s:

Area: A = s² (side squared)
Perimeter: P = 4s (four times the side)
Diagonal: d = s√2 (by Pythagoras' theorem)

Area Formula Explained

Area measures the amount of two-dimensional space enclosed by a shape. For a square with side length s, the area equals s multiplied by s, because length times width gives area for any rectangle, and a square is a rectangle where length and width are equal.

The result is expressed in square units. If the side is in metres, the area is in square metres (m²). If the side is in centimetres, the area is in square centimetres (cm²), and so on.

Diagonal Formula Explained

The diagonal of a square connects two opposite corners, cutting the square into two right-angled triangles. Each triangle has two shorter sides equal to the side length s and a hypotenuse equal to the diagonal d. Applying Pythagoras' theorem:

d² = s² + s² = 2s²

Therefore d = s√2. The constant √2 is approximately 1.41421. So a 5 m square has a diagonal of 5 x 1.41421 = 7.071 m.

Worked Example: Side of 5 m

Measurement	Formula	Calculation	Result
Area	A = s²	5 x 5	25 m²
Perimeter	P = 4s	4 x 5	20 m
Diagonal	d = s√2	5 x 1.41421	7.071 m

Common Side Lengths and Their Areas

Side length	Area	Perimeter	Diagonal
1 m	1 m²	4 m	1.414 m
2 m	4 m²	8 m	2.828 m
3 m	9 m²	12 m	4.243 m
5 m	25 m²	20 m	7.071 m
10 m	100 m²	40 m	14.142 m
100 m	10,000 m²	400 m	141.421 m

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Method: Standard Euclidean geometry. Area = s²; Perimeter = 4s; Diagonal = s√2, derived from Pythagoras' theorem (a² + b² = c² with a = b = s). All results are exact in rational arithmetic for integer inputs; the diagonal involves the irrational constant √2.

Results are mathematically exact for the inputs provided. Rounding in the displayed values is for readability only. When applying these results to physical measurements, allow for the precision of your original measurement.

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