Square
Definition, properties, construction, and formulas
Overview
A square is a regular quadrilateral with four equal sides and four right angles. It can also be seen as a rectangle with equal adjacent sides.
Core facts:
- Four sides of equal length
- All interior angles are 90°
- Both diagonals are equal and intersect at the center
- 4 axes of symmetry and 4-fold rotational symmetry
Additional property:
The area of the circumcircle is twice the area of the incircle.
Angle bisector
Median
Altitude and the Orthocenter
Triangle bisector
Pythagorean theorem
Pythagorean triples
Circle
Square
Rectangle
Rhombus
Parallelogram
Trapezoid
Construction with a Given Diagonal
Square Formulas
Area
Perimeter
Diagonal
Incircle radius
Circumradius
Key Points
- All square sides are equal: \(a = a = a = a\)
- Diagonals are equal and perpendicular bisectors of each other
- \(A = a^2\), \(U = 4a\), \(d = a\sqrt{2}\)
- One known measure is enough to derive all others
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