Square Calculator

Solve a square from any one measurement — side, area, perimeter or diagonal — with every step shown and the figure drawn to your numbers.

This is the geometry of one square. It does not raise a number to a power (x²), and it does not price floor space by the square foot — those are different questions with different answers.

Your square

I know the… Side Area Perimeter Diagonal

Side (a)

Change what you know above — the other three follow.

Unit

Decimals

Square with a side of 20 ft, drawn to scale

Area

400 ft²

A = a² = 20²

Side

20 ft

The value you entered

Perimeter

80 ft

P = 4a = 4 × 20

Diagonal

28.28 ft

d = a sqrt(2) = 20 × sqrt(2)

Step by step

You entered the side: a = 20 ft.
Area: A = a² = 20 ft² = 400 ft².
Perimeter: P = 4a = 4 × 20 ft = 80 ft.
Diagonal: d = a sqrt(2) = 20 ft × sqrt(2) = 28.28 ft.

Every line substitutes the number you typed, and rounds it the way the decimals control above is set.

Solve a square from any one measure

You know	Side	Area	Perimeter	Diagonal
Side (a)	a	a²	4a	a sqrt(2)
Area (A)	sqrt(A)	A	4 sqrt(A)	sqrt(2A)
Perimeter (P)	P / 4	P² / 16	P	P sqrt(2) / 4
Diagonal (d)	d / sqrt(2)	d² / 2	2d sqrt(2)	d

The highlighted row is the one you are using. Every row rebuilds the same square from a different starting measurement.

Every measure of this square

Measure	Value	Formula
Side (a)	20 ft	a
Area (A)	400 ft²	a²
Perimeter (P)	80 ft	4a
Diagonal (d)	28.28 ft	a sqrt(2)
Inradius (r)	10 ft	a / 2
Circumradius (R)	14.14 ft	a sqrt(2) / 2

The inradius is the circle that fits inside the square; the circumradius is the circle through its four corners. Both follow from the side.

This area in other units

Unit	Amount
Square centimeters (cm²)	371,612
Square meters (m²)	37.16
Square kilometers (km²)	0.0000372
Hectares (ha)	0.0037
Square inches (in²)	57,600
Square feet (ft²)	400
Square yards (yd²)	44.44
Acres	0.0092

Factors from NIST SP 811, all exact. These eight rows always use the automatic precision, whatever the decimals control says.

Formulas from Wolfram MathWorld; unit factors from NIST SP 811. Checked 7 September 2026.

Need a shape other than a square? The area calculator does twelve of them →

Square calculator. Area, perimeter, diagonal and side from any one measure.

A square calculator takes any one measure of a square — side, area, perimeter or diagonal — and returns the other three. Every measure follows from the side, so one number is enough: an area of 250 ft² comes back as a side of 15.81 ft, a perimeter of 63.25 ft and a diagonal of 22.36 ft.

The Four Measures of a Square, and Why Any One of Them Fixes the Rest

A square has a single free measurement. Fix the side and the other three are settled with it, which is why one number here fills four cards. Multiply the side by itself for the area, add the four equal sides for the perimeter, multiply the side by sqrt(2) for the corner-to-corner diagonal. A patio 20 ft on a side covers 400 ft², takes 80 ft of edging around it, and measures 28.28 ft across the middle.

That direction is mental arithmetic. The question that sends people searching runs the other way: the floor is 250 ft², so how long is one wall? A square root answers it — 15.81 ft, with a perimeter of 63.25 ft and a diagonal of 22.36 ft — and a square root is the step people come to check.

Seven of the ten results Bing returns for "area of a square" are school explainers rather than calculators, and they are built for the forward direction. Third Space Learning's US guide works six examples and every one of them turns a side into an area, from counting nine unit squares to a 4 ft wall covering 16 ft²; there is no calculator on the page and no route back from an area. Cuemath does carry the diagonal route (d²/2) and the perimeter route in its lesson, with three worked examples, then links out for the arithmetic. What none of them can do is take your number and show it substituted.

Area and perimeter count different things, and the units say so. Perimeter is a plain length in feet, meters or millimeters. Area is counted in squares of that unit, which is why it is written ft², m², mm²: the 400 ft² above is 400 one-foot squares, and the same patio reads 37.16 m², 44.44 yd² or 57,600 in² in the units table under the result. Because both dimensions grow together, doubling a side quadruples the surface — a 5 m square covers 25 m², a 10 m square covers 100 m² — while the edging only doubles.

Two different questions share this page's name. Raising a number to a power (x²) is algebra, and CalculatorSoup runs it as a separate page from its geometry one. "Square footage" is a property-and-flooring measurement that happens to be quoted in ft². Every field here is a length or an area belonging to one square figure. For a shape that is not a square, the area calculator covers twelve of them and runs the forward direction: dimensions in, area out.

Every Square Formula, and Where the sqrt(2) Comes From

A = a^2 \qquad P = 4a \qquad d = a\sqrt{2} \qquad r = \frac{a}{2} \qquad R = \frac{a\sqrt{2}}{2}

$a$ = Side length. The one measurement everything else is built from.
$A$ = Area, counted in squares of the unit you measured in: ft², m², cm².
$P$ = Perimeter, the distance once around the outside, in plain length units.
$d$ = Diagonal, corner to opposite corner across the middle.
$r$ = Inradius, the radius of the circle that fits inside and touches all four sides.
$R$ = Circumradius, the radius of the circle that passes through all four corners.

Two of the three headline formulas are counting. Area multiplies the side by itself; perimeter adds four equal sides. The diagonal is the one that needs proving, and Pythagoras does it in a line.

Cut the square along a diagonal and you are left with a right triangle whose two legs are both sides of the square, with the diagonal as the hypotenuse:

d^2 = a^2 + a^2 = 2a^2 \quad \Rightarrow \quad d = a\sqrt{2}

The multiplier is about 1.4142 and never settles into a repeating decimal. Wolfram MathWorld calls the diagonal of the unit square Pythagoras's constant, and the ratio holds for every square there is, in any unit: a 12 m square measures 16.97 m corner to corner.

Running the three formulas backwards is what the form exists for, and each one inverts on its own:

a = \sqrt{A} \qquad a = \frac{P}{4} \qquad a = \frac{d}{\sqrt{2}}

Squaring and taking a square root undo each other, which is the whole of the first one: if the area is the side multiplied by itself, then the number that multiplies by itself to give the area is the side. An area of 100 m² gives a side of 10 m, a perimeter of 40 m and a diagonal of 14.14 m.

Chain two inverses and you get the shortcuts homework questions ask for directly:

A = \frac{P^2}{16} \qquad A = \frac{d^2}{2}

A perimeter of 100 m is a side of 25 m, so the area is 625 m². A diagonal of 10 ft halves its square to 50 ft². The page never asks you to chain anything: it derives the side first, builds the other three from that, and prints each substitution with your own figure in it.

Two further measures sit in the table under the cards and on no card at all. MathWorld gives them in the general regular-polygon form, r = (1/2)a·cot(π/4) and R = (1/2)a·csc(π/4), which for four sides collapse to:

r = \frac{a}{2} \qquad R = \frac{a\sqrt{2}}{2} = \frac{d}{2}

The inradius is the largest circle you could cut from the square: 10 ft for a 20 ft square. The circumradius reaches the corners instead, 14.14 ft for the same one, and it is always half the diagonal.

Using the Form: Say What You Know, Then Type One Number

There is no Calculate button. The cards, the drawing, the step lines and the three tables all move while you type, and a copied link reopens the same square.

1. Pick your starting point on the I know the… bar: Side, Area, Perimeter or Diagonal.

2. Type the number. The field renames itself to match — Side (a), Area (A), Perimeter (P), Diagonal (d) — and the hint underneath reads "Change what you know above — the other three follow."

3. Set the Unit once. Feet is the default, followed by in, yd, mi, mm, cm, m and km, with No unit last. Starting from an area squares the suffix on the field for you, so it reads ft² instead of ft.

4. Leave Decimals on Automatic unless you want more digits. Automatic rounds the way a person writing it down would — whole numbers from 1,000 up, two decimals from 1 up, four below 1 — while 0, 2, 4 and 6 pin the count and pad it, so 4 turns 400 into 400.0000.

5. Read the Area card first. It is the largest and it always comes first, whichever measure you started from, with Side, Perimeter and Diagonal behind it. Each card carries its own substitution underneath, such as a = sqrt(A) = sqrt(250) on the side card; the card matching what you typed says "The value you entered" instead.

6. Look at the drawing beside the form. It is built from the figure you typed, with the side dimensions and the diagonal marked on the square itself; CalculatorSoup's square page illustrates the same geometry with one static SVG that looks identical whatever you enter. The proportions here stay honest because a square is 1:1 at any size.

7. The working is already open. Step by step ships expanded — four numbered lines with your own number substituted into each. For an area of 250 ft², the second reads "Side: a = sqrt(A) = sqrt(250 ft²) = 15.81 ft."

8. Three tables sit under that. Solve a square from any one measure highlights the row you are on, so the four formulas that produced your answers stay visible beside the twelve you are not using. Every measure of this square adds the inradius and the circumradius that no card carries. This area in other units is the only section that ships shut, and reads the area across eight units from cm² to acres.

9. Choose No unit for homework. Every unit string disappears from the cards, the steps and the drawing, the units table hides itself since there is nothing left to convert, and the arithmetic is untouched.

The formulas come from Wolfram MathWorld and the area factors from NIST Special Publication 811, both listed with the tool.

Four Squares, Read Off the Live Calculator

The page as it opens: a 20 ft patio

The calculator has answered before you touch it, on Side, 20, Feet (ft):

Area 400 ft², subtitled A = a² = 20²
Perimeter 80 ft, which is what edging is sold against
Diagonal 28.28 ft, corner to corner
Inradius 10 ft and circumradius 14.14 ft, in the measures table

Open This area in other units and the same patio reads 37.16 m², 44.44 yd², 57,600 in² and 371,612 cm², with square kilometers, hectares and acres on the rows you are not reading. The yd² row is the one a landscaper quotes from; the in² row is where a tile count starts.

Type 80 into Perimeter instead and nothing moves: the same 20 ft side, the same 400 ft², the same 28.28 ft diagonal. All four routes rebuild one square, which is the point of the "Solve a square from any one measure" table under the cards.

Backwards from a floor area: 250 ft² is a 15.81 ft wall

A studio listed at 250 square feet, a deck priced by the square foot, or a homework line that hands you the area and asks for the side — all the same run, on Area, 250, Feet (ft):

Side 15.81 ft, from a = sqrt(A) = sqrt(250)
Perimeter 63.25 ft
Diagonal 22.36 ft
Inradius 7.91 ft, circumradius 11.18 ft

The square root is why this direction needs a tool at all. sqrt(250) is irrational, so 15.81 ft is a rounded reading of a number that never ends, and squaring the rounded figure will not land back on exactly 250. Raise Decimals to 6 before copying it into anything you are about to cut.

Two pages that rank for this query show where the field stops. GIGAcalculator's area-of-a-square calculator takes a side and returns an area, with no perimeter, no diagonal and no way in from an area at all. AnalyzeMath's square solver does accept all four starting points, but carries no units anywhere and waits for a Calculate button before it answers.

From a perimeter, the exam classic: P = 100 m

"Find the area of a square whose perimeter is 100 m" is the shape of the question filling homework sites, and Bing suggests "area of a square given the perimeter" on its own. On Perimeter, 100, with the unit set to m:

Side 25 m, from a = P / 4 = 100 / 4
Area 625 m²
Diagonal 35.36 m

The one-step version is A = P²/16, and the trap is squaring the perimeter itself: 100² is 10,000, sixteen times the real answer. Dividing by four first keeps it straight, and that is the order the step lines print in.

Switch Unit to No unit and the same run gives 25, 625 and 35.36 with no unit strings anywhere, which is what a marked answer sheet wants.

From the diagonal, and the rounding it exposes: d = 10 ft

"Area of square with diagonal" is one of Bing's own suggestions under the diagonal query, and it is the route where the arithmetic is worth watching. On Diagonal, 10, Feet (ft):

Side 7.07 ft, from a = d / sqrt(2) = 10 / sqrt(2)
Area 50 ft²
Perimeter 28.28 ft
Circumradius 5 ft, always half the diagonal

By hand, A = d²/2 = 100/2 = 50 exactly. The calculator takes the longer road, dividing by sqrt(2) and squaring the result, and binary arithmetic cannot quite return to 50 from an irrational detour: the stored value is 49.99999999999999 and Automatic rounding prints 50. The artifact is a property of binary arithmetic, and the rounding rule is what keeps it off the screen.

Halving the diagonal is the mistake to avoid. The 5 ft is the circumradius, the reach from the center to a corner. The side is the diagonal divided by sqrt(2), about 0.7071 of it.

Nine Squares, Solved From Each of the Four Starting Points

What you type	Side	Area	Perimeter	Diagonal
Side = 20 ft	20 ft	400 ft²	80 ft	28.28 ft
Area = 250 ft²	15.81 ft	250 ft²	63.25 ft	22.36 ft
Perimeter = 80 ft	20 ft	400 ft²	80 ft	28.28 ft
Diagonal = 10 ft	7.07 ft	50 ft²	28.28 ft	10 ft
Side = 12 m	12 m	144 m²	48 m	16.97 m
Area = 100 m²	10 m	100 m²	40 m	14.14 m
Perimeter = 100 m	25 m	625 m²	100 m	35.36 m
Side = 5 m	5 m	25 m²	20 m	7.07 m
Area = 1 m²	1 m	1 m²	4 m	1.41 m

Where a Square Goes Wrong on Paper

Halving the diagonal to get the side. Half a diagonal is the circumradius, the distance from the center out to a corner. The side is the diagonal divided by sqrt(2), roughly 0.7071 of it, so a 10 ft diagonal gives a 7.07 ft side and not 5 ft.
Squaring the perimeter. The area from a perimeter is P²/16, never P². Divide by four to get the side, then square that: a perimeter of 100 m is a side of 25 m and an area of 625 m², where squaring 100 gives 10,000 and overshoots by a factor of sixteen.
Writing the area in plain feet. Area is counted in squares of the unit, so a 20 ft square is 400 ft². Choose Area as your starting point and the field suffix changes to ft² on its own, which is the reminder built into the form.
Doubling the side and expecting double the area. Both dimensions grow at once, so the area grows by the square of the factor. A 5 m square covers 25 m² and a 10 m square covers 100 m²: four times the surface for twice the edging.
Rounding at the first step. sqrt(250) is irrational, and the 15.81 ft on screen is a reading of it rather than the number itself. Carry the unrounded value into any arithmetic that follows, or set Decimals to 6 before copying.
Measuring one side and assuming the other three. Every figure here rests on the shape being a genuine square. Measure the adjacent side as well, and if the two differ, what you have is a rectangle — the area calculator is the one to open. Equal diagonals on their own prove nothing, since a rectangle has those too.

What This Page Will Not Do: x², Floor Pricing, Other Shapes

It is geometry, not algebra. Raising a number to a power (x²) shares the name and nothing else. Every field here is a length or an area belonging to one figure, and a value of zero or below goes nowhere: the four cards fall to dashes and a line names the measurement that has to be greater than zero.

It is not a square-footage estimator. A listing's square footage, a flooring quote and the waste allowance on top of it are a pricing job with a supplier's rules attached. This page measures one square figure and reports it in eight area units, with no price and no allowance in sight.

One shape only. A rectangle, a circle, a hexagon or a triangle belongs to the area calculator, which covers twelve figures and runs dimensions in, area out. The link at the foot of the tool goes straight there.

No exact radicals. The side of a 250 ft² square is sqrt(250), and the steps substitute it numerically as 15.81 ft rather than simplifying it. Six decimal places is as close to the exact value as the page gets.

Acres and hectares are output only. The eight-row table reads your area in them, but the selector on the form holds length units. For an area you already have in acres, hectares or square meters, the area converter turns it round.

Displayed rounding is not the stored value. Automatic rounding prints what a person would write down, while the full double-precision number sits behind it. That is why a 10 ft diagonal stores an area of 49.99999999999999 and shows 50, and why Decimals goes up to 6 for anyone who needs the digits.

The Words on the Cards and in the Tables

Side (a)

The length of one edge. A square has four of them and they are equal, which is what makes a single measurement enough to rebuild the whole figure.

Area (A)

The surface the square covers, equal to the side multiplied by itself. A 12 m square covers 144 m²; a 20 ft square covers 400 ft².

Perimeter (P)

The distance once around the outside, four times the side. It buys edging, fencing, framing and trim, none of which scale with the area: 20 ft of side is 80 ft of perimeter.

Diagonal (d)

The straight line from one corner to the opposite corner, equal to the side times sqrt(2). It is the longest distance inside the square, which is why it decides whether a square panel fits through a doorway.

Pythagoras's constant

sqrt(2), about 1.4142, the diagonal of a square whose side is 1. Wolfram MathWorld uses the name because the value falls straight out of the Pythagorean theorem applied to a square, and it is irrational: the decimals never repeat.

Inradius (r)

The radius of the circle that fits inside the square touching all four sides, which is half the side. For a 20 ft square it is 10 ft, and it is the largest circle you can cut from that square.

Circumradius (R)

The radius of the circle through all four corners, equal to half the diagonal. A 20 ft square has a circumradius of 14.14 ft, so a round table of that radius would just reach its corners.

Square unit

The unit an area is counted in, always the square of a length unit: ft², m², cm². One square meter is 10,000 cm² because a meter is 100 cm and 100 squared is 10,000, which is why the units table spans eight magnitudes.

Squares, Areas and Diagonals — Frequently Asked Questions

How do I find the side length of a square from its area?

Take the square root of the area: a = sqrt(A). An area of 250 ft² gives a side of 15.81 ft, and 100 m² gives exactly 10 m. Choose Area on the calculator, type the figure, and the side card shows the substitution it used.

What is the area of a square if I only know the perimeter?

Divide the perimeter by four to get the side, then square it, which is A = P²/16. A perimeter of 100 m is a 25 m side and an area of 625 m². Squaring the perimeter itself gives 10,000, sixteen times too much.

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

Halve the square of the diagonal: A = d²/2. A 10 ft diagonal gives 50 ft², along with a 7.07 ft side and a 28.28 ft perimeter. The calculator reaches the same answer the long way, dividing the diagonal by sqrt(2) to get the side first.

Why is the diagonal of a square the side times the square root of 2?

The diagonal cuts the square into two right triangles whose legs are both sides of the square. Pythagoras then gives d² = a² + a² = 2a², so d = a·sqrt(2), about 1.4142 sides. A 12 m square measures 16.97 m corner to corner, and a 20 ft square 28.28 ft. Wolfram MathWorld calls sqrt(2) Pythagoras's constant for this reason.

Is a square calculator the same as squaring a number, x²?

No. Squaring a number is algebra and applies to anything, including negatives. This page works on one geometric figure, so its fields are lengths and areas and it rejects anything at or below zero. CalculatorSoup runs both as separate tools for the same reason, and the plaque at the top of this one says which job it is doing.

Is this the same as a square footage calculator?

No. Square footage is a property and flooring measurement: the floor area of a room or a building, usually with a waste allowance and a price attached. This page measures a square figure and reports its area in eight units. For a room that is not a square, use the area calculator, which covers twelve shapes.

Why is the area written in square feet when the perimeter is in plain feet?

They measure different things. Perimeter is a distance, so it stays in feet. Area counts the squares that fit inside, so its unit is the square of a length: 400 ft² is 400 one-foot squares. Writing an area in feet is the most common slip in a marked answer.

What are the inradius and circumradius of a square?

The inradius is half the side, the radius of the circle that fits inside touching all four sides. The circumradius is half the diagonal, the radius of the circle through the four corners. A 20 ft square has 10 ft and 14.14 ft. Both appear in "Every measure of this square" under the cards, and they are what you need when a circle has to be cut from a square panel or a square has to sit inside a circle.

Can I use this calculator without units, for homework?

Yes. The unit selector ends with No unit, which strips every unit string from the cards, the steps and the drawing while leaving the arithmetic identical. The area conversion table hides itself, since there is nothing to convert. A perimeter of 100 still returns 25, 625 and 35.36.

How exact are the numbers this square calculator prints?

The formulas are exact and come from Wolfram MathWorld, with the area conversion factors taken from NIST Special Publication 811. What is rounded is the display: whole numbers from 1,000 up, two decimals from 1 up, four below 1, unless you fix the count with the Decimals control. Behind that sits full double precision, which is why a 10 ft diagonal holds an area of 49.99999999999999 and shows 50.

Does a square enclose the most area for a given perimeter?

Among four-sided figures, yes. Any rectangle with the same perimeter covers less, and the further from square it gets the worse it does. Across all shapes the circle wins, which is why a round pen holds more than a square one built from the same fencing.

I have the area — what do I actually order from it?

Split the order in two. Surface material follows the area: 400 ft² of paving, turf or tile, quoted as 44.44 yd² or 37.16 m² if that is the unit your supplier works in, and the units table has both. Edging, framing and trim follow the perimeter instead, 80 ft for the same square, and no amount of area tells you that number. Keep the diagonal as the fit check: it is the longest dimension the finished square has, so it decides whether the panel turns a corner or goes through a door.

Sources & References

Content verified by the Smart Calculators Team