Square ‒ Circumscribed, Inscribed Circle

A square is a basic geometric shape which has four sides of equal length and each internal angle is exactly 90 degrees. Each square has four vertices, our contains vertices A, B, C, D. In this case we speak of a square ABCD. These vertices are connected by lines to form four edges of a square. Specifically, the edges AB, BC, CD, DA. Each of these edges has the same length, namely the length a = 6. Each edge grips with its neighbouring edge right angle, i.e. the angle of 90 degrees.



**Diagonals of a square**

Green lines are marking the diagonals. Each square has two diagonals, this one has diagonals AC and DB. Diagonal is the line that connects two opposite vertices of the square. More facts about the diagonals:

- Diagonal is always longer than the edge of the square. More precisely: if the length of the edge of the square is a, then the diagonal has a length u = a.√2. That's just an application of the Pythagorean theorem.

- Diagonals always intersect in the center of the square (the center of gravity).

- Diagonal divides the square into two halves. The two diagonals divide the square into four quarters.

- The diagonals actually bisect each other. If you mark the center point of the square S (as shown), then the length of the segment AS will be the same as the length of the segment CS.

- Diagonal divides the angle between adjacent edges. For example, in the image, the angle ABC has the size of 90 degrees and the angle ABD has a size of 45 degrees.

- The diagonals form a right angle together.



**Circumference and Area**

The circumference is the total length of the edge of a square, therefore the sum of the lengths of all edges. So if a square has the length of an edge = a, then the circumference is equal to 4 times a. Area of a square is the space which it occupies. To calculate the area, take the length of one edge and multiply it by the length of the adjacent edge. But because the square has all edges the same length, you can just multiply a times a.



**Mathematical formulas**:



Circumference:

O = 4a



Area:

S = a²



Diagonal:

u₁ = a√2



The radius of the circle circumscribed:

rₒ = ½u₁



The radius of the circle inscribed:

rᵥ = ½a