Math
Square Calculator
A square is fixed by a single number, so the square calculator only needs one: side, diagonal, area, or perimeter.
It returns everything else, writes the diagonal exactly as a radical (side 5 gives 5√2), and adds the radius of the circle that fits inside and the circle that passes through the corners.
Area is in square units; the rest are lengths.
Pick the unit your measurements use. Area is reported in square units and volume in cubic units.
Try an example
Result
Area
—
- Side
- —
- Diagonal
- —
- Perimeter
- —
- Inscribed circle radius
- —
- Circumscribed circle radius
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the square calculator solves
A square is fixed by a single number, so the square calculator only needs one: side, diagonal, area, or perimeter. It returns everything else, writes the diagonal exactly as a A root expression written with the √ symbol, such as a square root. (side 5 gives 5√2), and adds the radius of the circle that fits inside and the circle that passes through the corners.
Formula
How to use it
- Choose the measurement you know and type its value.
- Pick a unit so the area gets a square-unit label.
- Read the side, diagonal (exact and decimal), area, perimeter, and circle radii.
How to read the answer
The diagonal of a square is always the side times √2 (about 1.4142). When the side is a whole number the calculator keeps that exact: side 5 → diagonal 5√2 ≈ 7.0711. Going the other way, a diagonal of 10 gives side 10/√2 = 5√2.
Common mistakes and edge cases
- Halving the diagonal to get the side — you must divide by √2, not 2.
- Squaring the perimeter instead of the side when finding area: A = (P/4)², not P².
- Forgetting that area is in square units while side, diagonal, and perimeter are plain lengths.
- A zero or negative value cannot be a square measurement and returns an error.
Worked examples
Side 5 (diagonal 5√2)
Area
25
Diagonal 10 (side 5√2)
Area
50
Area 49 in²
Area
49
Perimeter 26 ft
Area
42.25
Decimal side 3.5
Area
12.25
Area 50 (side 5√2)
Area
50
Negative side (impossible)
Area
Error
Frequently asked questions
What is the diagonal of a square formula?+
d = s√2. It comes from the Pythagorean theorem: d² = s² + s² = 2s². A square with side 5 has diagonal 5√2 ≈ 7.071.
How do I find the side of a square from its area?+
Take the square root: s = sqrt(A). Area 49 gives side 7; area 50 gives side √50 = 5√2.
How do I find the side from the diagonal?+
Divide by √2: s = d/√2 = d√2/2. A diagonal of 10 gives side 5√2 ≈ 7.071.
What is the radius of a circle inscribed in a square?+
Half the side: r = s/2. The circle through all four corners (circumscribed) has radius half the diagonal, R = s√2/2.
Is a square's perimeter related to its area?+
Yes, through the side: P = 4s and A = s², so A = (P/4)² and P = 4·sqrt(A). Perimeter 26 gives side 6.5 and area 42.25.
Why does the calculator show √2?+
Because s√2 is the exact diagonal. The decimal 7.0711 is rounded; exact radical form is what most geometry teachers want.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026