Square Calculator
Calculate the area, perimeter, and diagonal of a square using the side length.
Square Dimensions
Results:
Area: 0.00 square units
Perimeter: 0.00 units
Diagonal: 0.0000 units
Step-by-Step Solution:
About Squares
What is a Square?
A square is a regular quadrilateral with four equal sides and four right angles (90°). It is both a rectangle and a rhombus, combining the properties of both shapes.
Key Properties
- All four sides are equal in length
- All four angles are right angles (90°)
- Diagonals are equal, bisect each other at 90°, and bisect the angles
- Has four lines of symmetry (two diagonals and two midlines)
- Has rotational symmetry of order 4
- The diagonal divides the square into two congruent isosceles right triangles
Formulas
Area: A = s² (where s is the side length)
Perimeter: P = 4s
Diagonal: d = s√2 ≈ 1.414s
Side from Diagonal: s = d/√2 = d√2/2
Real-World Applications
- Games: Chess boards, checkers, Sudoku grids
- Construction: Tiles, floor patterns, ceiling panels
- Art: Canvas frames, pixel art, grid designs
- Technology: QR codes, app icons, pixel grids
- Architecture: Windows, building blocks, modular designs
Frequently Asked Questions
Why is the diagonal of a square s√2?
Using the Pythagorean theorem on the right triangle formed by two sides and the diagonal: d² = s² + s² = 2s², therefore d = √(2s²) = s√2.
Is a square also a rectangle?
Yes! A square is a special type of rectangle where all sides are equal. It has all the properties of a rectangle plus the additional property of equal sides.
How do I find the side length if I know the area?
Take the square root of the area: s = √A. For example, if the area is 64 square units, the side length is √64 = 8 units.
What's the difference between a square and a rhombus?
Both have four equal sides, but a square has all right angles (90°), while a rhombus can have any angles (as long as opposite angles are equal). A square is a special case of a rhombus.
How many lines of symmetry does a square have?
A square has 4 lines of symmetry: the two diagonals and the two lines connecting the midpoints of opposite sides.
Can I calculate perimeter from diagonal?
Yes! Since d = s√2, then s = d/√2. So perimeter = 4s = 4d/√2 = 2d√2 ≈ 2.828d.
What is the angle between the diagonals of a square?
The diagonals of a square intersect at 90° (right angles). They also bisect each other, meeting at the center of the square.
How is square area used in construction?
Square tiles are commonly used for flooring and walls. Knowing the area helps calculate how many tiles are needed, and the side length determines spacing and layout patterns.
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