A square has one measurement to give. Fix any single quantity — a side, a perimeter, a diagonal, an area, or the radius of either circle — and the other five follow, each by a fixed multiple or a fixed power of the side. That single degree of freedom is what this page is about, and it is why you can...
A square has one measurement to give. Fix any single quantity — a side, a perimeter, a diagonal, an area, or either of the two circle radii — and the other five follow, each by a fixed multiple or a fixed power. Enter whichever you have. Everything stays exact: a side of 5 has a diagonal of exactly 5√2, and 7.0711 is a rounding of it.
Any one of the six. Whole numbers, decimals and fractions all work.
SIDE — one edge
SIDE
5
Everything else on this page follows from this one number — a square has only one measurement to give.
| side | 5 | 5 |
| perimeter | 20 | 20 |
| diagonal | 5√2 | 7.071068 |
| area | 25 | 25 |
| inradius | 5/2 | 2.5 |
| circumradius | 5√2/2 | 3.535534 |
THE ANSWER PUT BACK
d² = 50 = 2 × 25 = 2s² ✓
R ÷ r = 1.41421356 → √2 ✓
PERIMETER
20
4s — no square root anywhere
DIAGONAL
5√2
s√2 — never whole when s is
AREA
25
s², one rung of the same ladder
INRADIUS
5/2
s ÷ 2 — the circle that fits inside
WHAT IS HAPPENING
d = 5 × √2 = 5√2
Always, for every square. That fixed ratio is why a square can never have its side and its diagonal both come out as whole numbers, or even both as fractions.
inside 5/2, through the corners 5√2/2
The inner circle touches the four sides and the outer passes through the four corners. Their radii are in the ratio √2, so the outer circle has exactly twice the area of the inner one.
How to double a square, and how not to. The obvious move is to double the side, and it is wrong: that gives four times the area, because area grows with the square of every length. The construction above is the right one, read backwards. The outer square is built on the inner square's diagonal, and it has exactly twice the area — which the fold demonstrates, since the four corner pieces that lie outside the inner square fold in and cover it exactly. To double a square, build on its diagonal; to halve one, join the midpoints of its sides.
Exact values throughout · surds kept as surds, never rounded away
Choose which measurement you have — side, perimeter, diagonal, area, inradius or circumradius — and type it in. Fractions such as 3/4 and decimals both work.
Read the side first, exactly. Every other quantity on the page is derived from it, so if the side carries a √2 then several of the others will too.
Check the ladder figure. Each bar is drawn to the same scale, so you can see that the diagonal really is 1.4142 times the side rather than being told so.
Press replay on the fold. The four corner pieces rotate into the inner square and cover it exactly — that is the halving construction, and read backwards it is how to double a square.
Take a square with a side of 5. The perimeter is 4 × 5 = 20. The area is 5² = 25. The inradius — the circle that fits inside, touching all four sides — is half the side, so 2.5. The diagonal is 5√2, which is about 7.0711. It is exactly 5√2 and it is not exactly 7.0711, and the difference matters as soon as the number feeds into anything else. The circumradius, the circle through the four corners, is half of that: 5√2/2, about 3.5355. Now walk in from the other end. Given a diagonal of 10, divide by √2 to get the side: 10 ÷ √2 = 5√2, about 7.0711. Note what happened — the √2 moved from the diagonal to the side and did not disappear. The area is then 50, which is a whole number again, because squaring the side cancels the root. That pattern is worth keeping. Given an area of 2, the side is √2, the diagonal is exactly 2, and the perimeter is 4√2. Given an area of 25, everything is tidy. The root appears and vanishes depending on which end you start from, and it can never be absent from both ends at once. Here is why. Suppose a square had a whole-number side and a whole-number diagonal. Then √2 = d ÷ s would be a ratio of two whole numbers, and √2 is not such a ratio. The same argument rules out fractions, since a ratio of fractions is still a ratio of whole numbers. Over the first two thousand whole-number sides, not one has a whole-number diagonal; over the first two thousand whole-number diagonals, not one has a whole-number side. Finally, doubling. A square of side 5 has area 25; a square of side 10 has area 100 — four times, not twice. To double the area you scale every length by √2, so the side becomes 5√2 ≈ 7.07. There is an older way to say the same thing without arithmetic: build the new square on the diagonal of the old one. The diagonal of the side-5 square is 5√2, and a square on it has area 50. The construction on this page shows it in reverse. Join the midpoints of a square's sides and the inner square has exactly half the area, because the four corner triangles left outside fold inward and cover the inner square precisely. Nothing is added and nothing is taken away, so the inner square is half — and therefore the outer square, which stands on the inner one's diagonal, is double.
| Rule | What it says | Why |
|---|---|---|
| One measurement is enough | any one fixes all six | A square has a single degree of freedom. Everything else is a fixed multiple or power. |
| Perimeter | P = 4s | The only conversion here with no square root in it, either way. |
| Diagonal | d = s√2 | The diagonal cuts the square into two right triangles with the sides as legs. |
| Back from the diagonal | s = d√2 ÷ 2 | Dividing by √2 leaves one behind — it cannot be cancelled away. |
| Never both rational | s and d | If both were fractions then √2 = d/s would be one, and it is not. |
| Inradius | r = s ÷ 2 | The circle that fits inside touches all four sides, so its diameter is the side. |
| Circumradius | R = s√2 ÷ 2 | The circle through the corners has the diagonal as its diameter. |
| The two circles | R ÷ r = √2 | So the outer circle has exactly twice the area of the inner one. |
| Square inside a circle | 2 ÷ π ≈ 63.66% | The share of the circumcircle the square actually covers, at any size. |
| Halving a square | join the midpoints | The square through the four midpoints has exactly half the area. |
| Doubling a square | build on the diagonal | Not on double the side — that gives four times, which is the classic trap. |
| The scale that doubles | √2 ≈ 1.414214 | Scaling every length by √2 doubles the area, because area goes as the square. |
| Rounding √2 | 1.41 is out by 0.4 in 141 | On a side of 100 that is nearly half a unit of diagonal. |
| A square is a rectangle | and a rhombus, and a kite | Four right angles and four equal sides at once, so it satisfies all three. |
All of the others. A square has a single degree of freedom, so any one quantity fixes the shape completely. From the side: the perimeter is 4s, the diagonal is s√2, the area is s², the inradius is s/2 and the circumradius is s√2/2. Every one of those runs backwards too, so you can start from whichever measurement you actually have.
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Last updated: September 1, 2026 · Exact values throughout, surds kept as surds · One measurement fixes a square completely.