Find the area, perimeter, diagonal, and side length of any square. Enter any one known value — the calculator instantly computes all other properties with full formula working shown.
✓Last verified: April 2026 · Standard geometry formulas
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Please enter a positive value.
The length of one side of the square
Area
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Area
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Side
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Perimeter
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Diagonal
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Sources & Methodology
✓All square property formulas are standard Euclidean geometry, consistent with definitions in Common Core State Standards for Mathematics (CCSS.MATH.CONTENT.6.G.A.1) and the NCTM Principles and Standards for School Mathematics.
US Common Core Standard 6.G.A.1 defines finding area of polygons including squares by composing and decomposing, forming the curriculum basis for the formulas in this calculator.
NCTM standards establish the expected knowledge of square geometry properties including the diagonal formula derived from the Pythagorean theorem.
Formulas: Area = s^2 | Perimeter = 4s | Diagonal = s x sqrt(2) | From area: s = sqrt(A) | From perimeter: s = P/4 | From diagonal: s = d/sqrt(2). Diagonal formula derived from Pythagorean theorem: d^2 = s^2 + s^2 = 2s^2, therefore d = s*sqrt(2). All calculations use double-precision floating point arithmetic.
⏱ Last reviewed: April 2026
Square Formulas: Area, Perimeter, Diagonal & Side — Complete Guide
A square is a regular polygon with four equal sides and four right angles. Because all four sides are identical, knowing just one measurement — side length, area, perimeter, or diagonal — is enough to calculate every other property. This makes squares one of the most straightforward shapes in geometry, and square calculations appear constantly in construction, flooring, land measurement, and everyday problem-solving.
All Four Square Formulas
Area = s² Perimeter = 4s Diagonal = s × √2 Side = √Area = P/4 = d/√2
From side (s = 7): Area = 7² = 49 | Perimeter = 4 × 7 = 28 | Diagonal = 7 × 1.41421 = 9.899 From area (A = 64): Side = √64 = 8 | Perimeter = 32 | Diagonal = 8 × 1.41421 = 11.314 From perimeter (P = 40): Side = 40/4 = 10 | Area = 100 | Diagonal = 14.142 From diagonal (d = 10): Side = 10/1.41421 = 7.071 | Area = 50 | Perimeter = 28.284
Quick Reference Table — Common Square Sizes
Side
Area
Perimeter
Diagonal
1
1
4
1.414
2
4
8
2.828
5
25
20
7.071
10
100
40
14.142
12
144
48
16.971
15
225
60
21.213
20
400
80
28.284
100
10,000
400
141.421
Why the Diagonal Formula Uses √2
The diagonal of a square divides it into two right triangles, each with two legs equal to the side length. Applying the Pythagorean theorem: d² = s² + s² = 2s². Taking the square root of both sides: d = s√2. Since √2 ≈ 1.41421, the diagonal of any square is always approximately 1.414 times its side length. This ratio is constant for all squares regardless of size, making it easy to estimate diagonals mentally once you know the side length.
Real-World Applications of Square Calculations
Square area calculations are used constantly in home improvement and construction. Calculating how much flooring, carpet, or tile to buy requires knowing the room area in square feet or square meters. A 12 ft x 12 ft room is a square with area = 144 ft². If tiles are sold per square meter, convert: 144 ft² = 144 / 10.764 = 13.38 m². Perimeter calculations tell you how much baseboard, fencing, or edging material to buy. Diagonal calculations are critical for checking whether a square room or frame is truly square — if the two diagonals are equal in length, the corners are at 90 degrees.
💡 The 3-4-5 check: To verify a corner is exactly 90 degrees (square), measure 3 units along one wall and 4 units along the adjacent wall. The diagonal between those points should be exactly 5 units (the 3-4-5 right triangle). For larger squares, multiply: 6-8-10, 9-12-15. This technique is used by builders and carpenters to ensure square corners without a large set square.
Frequently Asked Questions
Area = side x side = s squared. If the side is 5 units, area = 25 square units. If the side is 8 cm, area = 64 cm squared. The area unit is always the square of the length unit. To find the side from area: side = square root of area.
Side = square root of Area. If area = 64, side = sqrt(64) = 8. If area = 50, side = sqrt(50) = 7.071. If area = 144, side = sqrt(144) = 12. For areas that are not perfect squares, you get an irrational number. Use a calculator or this tool for exact results.
Perimeter = 4 x side. Since all four sides are equal, multiply one side by 4. A square with side 9 has perimeter = 36. To find the side from perimeter: side = perimeter / 4. If perimeter = 40, side = 10. If perimeter = 28, side = 7.
Diagonal = side x sqrt(2) = side x 1.41421. From the Pythagorean theorem: diagonal squared = side squared + side squared = 2 x side squared. So diagonal = side x sqrt(2). Example: side = 10, diagonal = 14.142. Side = 7, diagonal = 9.899. The diagonal is always about 41.4% longer than the side.
Side = diagonal / sqrt(2) = diagonal x 0.70711. Example: diagonal = 10, side = 10 / 1.41421 = 7.071. Diagonal = 20, side = 14.142. You can also compute: side = sqrt(diagonal squared / 2). Both methods give the same result.
Area = diagonal squared / 2. Since diagonal = side x sqrt(2), diagonal squared = 2 x side squared = 2 x area. Therefore area = diagonal squared / 2. Example: diagonal = 10, area = 100 / 2 = 50. Diagonal = 14.142, area = 200 / 2 = 100. This lets you find area directly from the diagonal.
First find side = sqrt(area). Then perimeter = 4 x sqrt(area). Example: area = 25, side = 5, perimeter = 20. Area = 100, side = 10, perimeter = 40. Area = 36, side = 6, perimeter = 24. The perimeter is always 4 times the square root of the area.
A 10 x 10 square has area = 10 x 10 = 100 square feet. Perimeter = 4 x 10 = 40 feet. Diagonal = 10 x 1.41421 = 14.14 feet. This is a common room or plot size. To convert: 100 ft squared = 9.29 m squared = 11.11 yards squared.
A 5 x 5 square has area = 25 square meters. Perimeter = 20 meters. Diagonal = 7.071 meters. In square feet: 25 m squared = 25 x 10.7639 = 269.10 ft squared. In square centimeters: 25 m squared = 250,000 cm squared.
A square has: 4 equal sides, 4 right angles (90 degrees each), 4 lines of symmetry (2 through opposite sides, 2 through opposite corners), 2 equal diagonals that bisect each other at 90 degrees, interior angles summing to 360 degrees. A square is both a special rectangle (all angles 90 degrees) and a special rhombus (all sides equal).
The area unit is the square of the side unit. Side in cm gives area in cm squared. Side in m gives area in m squared. Conversions: 1 m squared = 10,000 cm squared. 1 ft squared = 144 in squared. 1 yard squared = 9 ft squared. 1 acre = 43,560 ft squared. Enter the side in your preferred unit and the area is automatically in that unit squared.