The Rectangle

A rectangle is a parallelogram whose interior angles are all 90 degrees - a right angle. The opposite sides of a rectangle are always the same size. A square, then, is a special case of a rectangle that has all sides the same length.

Basic description

First, look at the picture: In the picture, you see a rectangle that is formed by the vertices A, B, C, and D, so it is a rectangle ABCD. It has four sides: AB, BC, CD, and DA. The sides opposite each other are always the same length, denoted by a and b. In the figure, the sides are five and three in length, as shown.

If the lengths of all the sides were equal, i.e. a = b, then it would also be a rectangle, but more often we call such a rectangle a square. Thus a square is just a special case of a rectangle.

Diagonals

Every rectangle has two diagonals, which are lines joining non-adjacent vertices. In our figure, these are the segments AC and BD. Also labeled u₁ and u₂ in the figure. These diagonals are always the same size. They are also always longer than either side of the rectangle. Other properties of diagonals:

• The length of a diagonal is equal to $|u|=\sqrt{a^2+b^2}$, by Pythagoras' theorem.
• Unlike a square, diagonals do not form right angles to each other.
• A diagonal divides a given rectangle into two halves. The two diagonals then divide the rectangle into four quarters.
• The diagonals themselves bisect each other. If we mark the centre of the rectangles with the point S (as in the figure), then the length of the line segment AS will be the same as the length of the line segment CS.

Perimeter and content

The perimeter is the length of the edge of the rectangle, that is, the sum of the lengths of all four sides: a + b + a + b. But since two opposite sides are always the same length, we can calculate the perimeter as 2 · a + 2 · b.

The area of a rectangle is the size of the area occupied by the rectangle. We calculate it by multiplying the length of one side by the length of the other, adjacent, side. Thus, the area of the rectangle is equal to a · b. Once again, it is clear:

$$\begin{eqnarray} (\mbox{ Circuit })\quad o &=&2\cdot a+2\cdot b\\ (\mbox{ Content })\quad S&=&a\cdot b \end{eqnarray}$$

The following figure shows first the perimeter - the sum of the lengths of the lines in red, and then the content - the coloured part of the rectangle.

The circle traced and inscribed

Similar to a square, a rectangle has a circumscribed circle, which is a circle that is centered at the midpoint (center of gravity) of the rectangle and has a diameter half the length of the diagonal. The circle circumscribed passes through all the vertices of the rectangle. Unlike a square, however, a rectangle does not have an inscribed circle; except, of course, when the rectangle is also a square.