From a side of 5 cm
- Perimeter: P = 4s = 4(5) = 20 cm.
- Area: A = s² = 5² = 25 cm².
- Diagonal: d = s√2 = 5√2 ≈ 7.071 cm.
- Radii: r = 5/2 = 2.5 cm; R = 5√2/2 ≈ 3.536 cm.
Square Calculator: Area, Perimeter, Diagonal & Side
Enter any known measurement of a square to calculate its side, area, perimeter, diagonal, inradius, and circumradius. You’ll get exact values where practical, rounded answers, and steps matched to your starting measurement.
Choose what you know, enter one positive value, and the results update immediately.
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Square measurements
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Results
Enter one known measurement to calculate the square.
Calculation steps
Square formulas and inverse formulas
Every result can be found by first recovering the side s. These direct forms are useful when your known measurement is not the side.
| Known value | Find the side | Useful direct formulas |
|---|---|---|
| Side s | s = s | P = 4s; A = s²; d = s√2; r = s/2; R = s/√2 |
| Area A | s = √A | d = √(2A) |
| Perimeter P | s = P/4 | A = P²/16; d = P√2/4 |
| Diagonal d | s = d/√2 | A = d²/2; R = d/2 |
| Inradius r | s = 2r | A = 4r²; d = 2r√2 |
| Circumradius R | s = R√2 | d = 2R; A = 2R² |
The diagonal formula follows from the Pythagorean theorem: d² = s² + s² = 2s². The inradius reaches a side midpoint, while the circumradius reaches a corner; therefore 2r = s and d = 2r√2.
Worked square examples
From a diagonal of 12 cm
- Side: s = d/√2 = 12/√2 = 6√2 ≈ 8.485 cm.
- Area: A = d²/2 = 12²/2 = 72 cm².
- Perimeter: P = 4s = 24√2 ≈ 33.941 cm.
- Radii: r = 3√2 ≈ 4.243 cm; R = d/2 = 6 cm.
From an area of 100 m²
- Side: s = √A = √100 = 10 m.
- Perimeter: P = 4s = 4(10) = 40 m.
- Diagonal: d = √(2A) = √200 = 10√2 ≈ 14.142 m.
- Radii: r = 5 m; R = 5√2 ≈ 7.071 m.
Square calculator FAQs
How do I find the area of a square from its diagonal?
Use A = d²/2. For a diagonal of 10 cm, A = 10²/2 = 50 cm².
How do I find the side of a square from its area?
Take the positive square root: s = √A. If A = 81 cm², then s = √81 = 9 cm.
How do I find the side of a square from its perimeter?
Divide the perimeter by four: s = P/4. A perimeter of 24 m gives s = 24/4 = 6 m.
How do I find a square's diagonal from its area or perimeter?
From area, use d = √(2A); A = 50 cm² gives d = 10 cm. From perimeter, use d = P√2/4; P = 20 cm gives d = 5√2 ≈ 7.071 cm.
What is the difference between inradius and circumradius?
The inradius r runs from the center to a side, so r = s/2. The circumradius R runs from the center to a corner, so R = d/2 = s/√2.
Why does square area use squared units?
Area multiplies one length by another: A = s × s. Thus 5 cm × 5 cm = 25 cm², not 25 cm.
Can a square have different length and width?
No. A square has four equal sides, so its length and width are equal. A four-right-angle shape with unequal length and width is a rectangle, not a square.
How does the calculator round answers?
Calculations retain full JavaScript numeric precision. Results show an exact fraction or radical when practical, plus a decimal rounded to the selected 0–12 places.