Square Area Calculator

Side (a)

Diagonal (d)

Perimeter (P)

Area (A)

Add a border or frame around the square

Enter a border width to get the framed outer size and the border-only area, for matting, edging, or a path around a square.

Estimate a tiling or material cost

Enter a price per square metre to estimate the cost of tiling, flooring, or painting the area.

Enter any one measurement, the side, diagonal, perimeter, or area, and the calculator fills in the other three for you.

Please input values into the fields

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Show how the area scales with the side

A bar chart of how the area changes as the side grows.

This square area calculator can not only find the area of a square. If you input one value for any of the following - side length, diagonal, perimeter or area - it will calculate the remaining three values. After entering a value and selecting a unit, the other results are automatically displayed with the area as the main result. You can also activate the "frame" option to find the dimensions of frames or openings, or the "costs" option to calculate tile work or flooring prices.

What is the area of a square?

A square is a quadrilateral with four equal sides and four right angles. Area is the size of the surface of a two-dimensional figure. Because this space is two-dimensional, it is measured in square units such as square inches or square feet. A square is determined by one value - its side length. So if you know one value, then all other values can be calculated.

How to use this calculator:

Enter the existing measurement (side length, diagonal, perimeter or area) into its corresponding field and select a unit such as centimeters or inches. After entering a value, the tool will automatically calculate the remaining three values and display the area as the main result. It is not necessary to specify which type of measurement was entered. To start over with another value, simply clear the previously filled-in field and enter a new value. By activating the "Frame" option, you can determine the dimensions of frames or openings, while activating the "Costs" option allows for an estimate of tile work or flooring prices.

Formula for calculating the area of a square:

The area is determined only by the length of the sides. You multiply the side length by itself, that is to say you square it.

A = a \times a = a^2

If instead you know the diagonal, then area is equal to the square of the diagonal divided by two. This is because the diagonal of a square is equal to its side length multiplied by the square root of 2.

A = \frac{d^2}{2}

If you know the perimeter, divide it by 4 to get the length of each side, then square that number.

A = \left(\frac{P}{4}\right)^2 = \frac{P^2}{16}

Assume that a square has sides of length 5 cm. The area can be calculated directly by squaring this value.

A = 5^2 = 25 \ \text{cm}^2

Conversely, if you know the area, you can find the side length by taking the square root of the area.

a = \sqrt{A}

Calculating the area of a square by its diagonal.

The diagonal connects two opposite corners and divides a square into two right triangles. According to the Pythagorean theorem, the length of the diagonal is equal to the side length multiplied by the square root of 2, so the square of the diagonal is twice the area of the square. This explains why the formula for the diagonal involves dividing by 2.

d = a\sqrt{2} \quad\Longrightarrow\quad A = \frac{d^2}{2}

If a square has a diagonal of 8 centimeters, its area is 32 square centimeters, since the formula for the area of a square given its diagonal is diagonal squared divided by 2. This calculator will allow you to simply type in the value of the diagonal (8) into the corresponding field and it will calculate the area for you.

Calculating area of a square from perimeter.

The perimeter is the total distance around a square. Since all four sides are equal length, the perimeter is simply four times the side length. Divide the perimeter by 4 to find the length of one side, then square that number. If the perimeter of a square is 52 units, then the length of each side is 13, and the area would be 169 square units. Enter the perimeter into the appropriate box below to instantly find the side length, diagonal, and area.

Check all measurements of a square:

Every measurement of a square is connected to the other measurements through its side length. The following table shows the relationships used in the calculator so that by entering one value you can calculate all the others.

Measurement	Symbol	Formula from the side
Diagonal	d	d = a times square root of 2
Perimeter	P	P = 4a
Area	A	A = a squared
Inscribed circle radius	r	r = a / 2
Circumscribed circle radius	R	R = a times square root of 2, over 2
Side from area	a	a = square root of A

Example calculation:

The following table shows the area, perimeter and diagonal for various typical side lengths. Note that the area increases very quickly as the side length gets larger because it is squared.

Side	Perimeter	Diagonal	Area
3 cm	12 cm	4.24 cm	9 sq cm
5 cm	20 cm	7.07 cm	25 sq cm
10 cm	40 cm	14.14 cm	100 sq cm
20 cm	80 cm	28.28 cm	400 sq cm

Why squaring the side length is important:

Because the length of a side is squared, doubling the length of a side does not double the area; it quadruples it. Tripling the length of a side will make the area nine times larger. This is important to remember when working with plans that are scaled up or down proportionally. A patio whose width has been doubled requires four times as much paving material, and a photo with twice the length of sides needs an area four times larger for printing. The zoom graph in the analysis section shows this relationship visually.

\frac{A(ka)}{A(a)} = \frac{(ka)^2}{a^2} = k^2

Comparison of squares, rectangles and rhombuses:

Since squares are special cases of other geometric shapes, the formulas for those shapes can be simplified to the formula for a square. The area of a rectangle is the product of its length and width. Since a square is a rectangle with equal length and width, the formula becomes the side length squared. The area of a rhombus is half the product of its two diagonals. Since a square is a rhombus with equal and perpendicular diagonals, its area is also half the square of the diagonal. If the sides of a shape are not all the same length or the angles are not right angles, use the rectangle or rhombus calculator.

Incircle and circumcircle

Each square has two associated circles. The inscribed circle lies inside the square and touches the midpoint of each side. Its radius is half the length of a side. The circumscribed circle passes through all four corners, and its radius is half the diagonal. In addition to the main results, both radii are shown in the calculation tool.

r_{\text{in}} = \frac{a}{2} \qquad R_{\text{out}} = \frac{d}{2} = \frac{a\sqrt{2}}{2}

Frames, borders and paths

In many real-world construction projects, a border is added around the square. Examples include the frame of a picture, the path around a lawn, or the decoration around a tile. If you turn on the "Border" option and enter the width of the border, then both the outer dimensions with the border and the area of the border itself are calculated - that is, the difference between the outer square and the inner one. If a square with side length a has a border of width b, then the area of the border is:

A_{\text{border}} = (a + 2b)^2 - a^2 = 4b\,(a + b)

Estimate costs for tiles or flooring

The area is used as a basis for estimating the cost of tiling, flooring, turfing or painting work. Once you know the area, you can multiply it by the price per unit area to get the total cost. You can check your estimate if you enable the "Cost" option and enter the price per square meter. Since the costs are calculated in square meters, you need to convert the side length or area into metric units to get the corresponding total amount. When purchasing materials, it is also important to plan for some wastage and cutting allowance.

Areas of application for calculating the area of squares:

Because squares are everywhere, this calculation is used in many different professions. Construction workers and landscapers determine the dimensions of patios, planks, lawns, and square pavement areas. Tile setters and floor layers calculate how many square tiles or flooring pieces they need for a room. Picture frame makers and artists cut out square sheets and canvases. Buying and selling land is done by calculating its area, usually in square meters or feet. Even board games like chess and logic puzzles are made up of equal-sized squares.

Tips for accurate results:

Measure one side from corner to corner and do this twice. Small errors in the length of a side can lead to large errors in area when squared. Make sure your units match up. Since a length in centimeters will give an area in square centimeters, convert your units before comparing values. Finally, make sure that the figure is actually a square with four equal sides and four right angles. If the sides are not all the same or if the corners do not have right angles, use the formula for rectangles instead.

Frequently asked questions

What is the formula for finding the area of a square?

If the area is denoted by A, it is equal to the square of the length of a side. So for example, if a square has sides of 5 units in length, the area is 25 square units. All four sides are of equal length and all four angles are right angles, so knowing just the length of one side is sufficient.

How do you find the area of a square using its diagonal?

The value used is the square of the diagonal divided by two. Since a square's diagonal is equal to its side length multiplied by the square root of 2, the square of the diagonal is twice as large as the area. If the diagonal is 8, then the area is 32 units squared, which is equivalent to 64 divided by 2. The area can be easily calculated by entering the length of the diagonal into the appropriate field.

How do you find the area of a square given its perimeter?

Divide the perimeter by four to obtain the length of a side, then square that length. If the perimeter is 52 units, the length of a side is 13 and the area is 169 square units. In this calculator, typing the perimeter fills in the side, diagonal, and area at once.

What happens to area when you double the side length?

The area does not double but quadruples. This is because the area is determined by the square of the length of the sides. If you triple the length of the sides, the area will be nine times larger, which explains why the cost for tiling or paving when doubling the width would be about four times higher.

What is the difference between area and perimeter of a square?

The area is the space inside of the square and is expressed in square units such as square meters. The perimeter is the length around the outside edge and is expressed in regular units such as meters. A square with a side length of 5 has an area of 25 square units while its perimeter is 20 units.

Is it possible to calculate the length of a side if only the area is known?

Yes. You take the square root of the area. The side length is the square root of A. If the area is 144, then the side length is 12. If you enter the area in the appropriate field, this calculator will immediately perform the calculations and also give you the perimeter and diagonal.

Related calculators

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Area CalculatorFind the area of a square, rectangle, triangle, circle, sector, ellipse, trapezoid, parallelogram, or rhombus. Pick a shape, enter its measurements, and get the area and perimeter in any unit.

Circle Area CalculatorEnter any one of a circle's radius, diameter, circumference, or area and get the other three instantly, plus an optional sector area and arc length.

Triangle Area Calculator (3 Sides)Find the area of a triangle from its three side lengths with Heron's formula, or enter the area with two sides to solve for the third. Also gives the angles, heights, medians, and inscribed and circumscribed circles.

Rectangle Area CalculatorEnter any two of a rectangle's length, width, area, diagonal, or perimeter and get all the others instantly, with full unit conversion and an optional cost estimate.

Hexagon Area CalculatorEnter any one of a regular hexagon's side, perimeter, long or short diagonal, circumradius, apothem, or area and get all the others instantly, with full unit conversion.

Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Square
Definition and properties of the square, including its diagonal and area.
Wikipedia: Area
How area is defined and measured for plane shapes.
Math is Fun: Square
Plain-language explanation of a square's sides, diagonal, perimeter, and area.