Square ‒ Circumscribed, Inscribed Circle

A square is a basic geometric shape with four sides of equal length, and each internal angle is exactly 90 degrees. A square has four vertices; in this example, we denote them A, B, C and D, forming square ABCD. These vertices are connected by four edges: AB, BC, CD and DA, each of equal length, specifically a = 6. Each edge meets its adjacent edge at a right angle, meaning all internal angles are 90 degrees. **Diagonals of a Square** The green lines represent the diagonals of the square. Every square has two diagonals, and in this case, they are AC and BD. A diagonal connects two opposite vertices of the square. Some important facts about diagonals include: - A diagonal is always longer than the edge of the square. Specifically, if the length of the edge is a, then the diagonal has a length u = a.√2, as derived from the Pythagorean theorem. - The diagonals intersect at the center of the square, which is also the center of gravity. - Each diagonal divides the square into two equal halves, and together, the two diagonals divide the square into four equal quarters. - The diagonals bisect each other, meaning the center point of the square, labeled S, divides each diagonal into two equal parts. For example, the lengths of segments AS and CS are equal. - A diagonal bisects the angle formed by two adjacent edges. In this case, the angle ABC is 90 degrees, and diagonal BD divides this angle into two 45-degree angles. - The two diagonals intersect at a right angle (90 degrees). **Perimeter and Area** The perimeter of a square is the total length of its edges. Since all four edges have equal length, the perimeter is simply four times the length of one edge: P = 4a. The area of a square represents the space it occupies. To calculate the area of a square, multiply the length of one edge by itself: A = a². **Mathematical formulas**: Perimeter: P = 4a Area: A = a² Diagonal: u₁ = a√2 Radius of the circumscribed circle: rₒ = ½u₁ Radius of the inscribed circle: rᵥ = ½a