Čtverec

A square is a basic geometric shape with four equal sides, and each internal angle measuring exactly 90 degrees. There is also a method called completing the square.

Popis

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Every square consists of four vertices; ours includes vertices A, B, C, D, which are marked in blue in the diagram. We refer to it as square ABCD. These vertices are connected by line segments that form the four sides of the square, specifically: AB, BC, CD, DA. In the diagram, these sides are indicated by bold lines. Each of these sides is the same length, specifically five units.

Each side forms a right angle with its adjacent sides, that is, an angle of 90 degrees.

Úhlopříčky čtverce

The green lines represent the diagonals. Every square has two diagonals; this one has diagonals AC and DB. A diagonal is thus a line segment that connects two opposite vertices of the square. Additional facts about diagonals:

• The diagonal e is always longer than the side of the square.
• Přesněji: pokud má strana čtverce délku a, pak úhlopříčka u má délku $|u|=a\cdot\sqrt{2}$. Je to jen aplikace Pythagorovy věty.
• The diagonals always intersect at the center of the square (at the centroid).
• The diagonal divides the given square into two halves. Both diagonals then divide the square into four quarters.
• The diagonals bisect each other. If we mark the center of the square with point S (as in the diagram), then the length of segment AS will be the same as the length of segment CS.
• The diagonal divides the angle between adjacent sides. For example, in the image, angle ABC is 90 degrees and angle ABD is 45 degrees.
• The diagonals intersect at a right angle.

Obvod a obsah

The perimeter is the length of the boundary of a square, that is, the sum of the lengths of all its sides. Therefore, if a square has sides of length a, then the perimeter is 4 · a. The area measures the size of the surface covered by the square. Take the length of one side and multiply it by the length of an adjacent side. However, since all sides of a square are of equal length, you can simply multiply a · a. To summarize again:

$$\begin{eqnarray} (\mbox{ obvod })\quad o &=&4\cdot a\\ (\mbox{ obsah })\quad S&=&a\cdot a=a^2 \end{eqnarray}$$

For example, the square in the previous image has a perimeter of 4 · 5 = 20 and an area of 5 · 5 = 25.

Poloha čtverce

Is the following figure a square?

The previous figure is a square because it has four sides, all of equal length, and each side forms a right angle with its adjacent sides. If we rotate the square, it remains a square. However, if we change the sizes of the angles, it will no longer be a square:

Four vertices, four sides of equal length, but not all angles are right angles, so it is not a square.

Jak narýsovat čtverec

Drawing a square is straightforward. All you need to know is the length of the side. If the side length is, for example, three centimeters, you first draw a segment three centimeters long. Label the endpoints as vertices A and B (or whatever you prefer). Then, using a ruler, measure a right angle and draw two perpendicular segments from the points A and B in the direction you choose. These segments will again be three centimeters long. Label the new points C and D. Finally, connect the points C and D with a segment.

Kružnice opsaná a vepsaná

The circumscribed and inscribed circles are two concepts that occur along with the square. Both circles have their center at the center of the square, which is at the intersection of the diagonals.

The circumscribed circle is a circle that intersects all the vertices of a square. It has a radius AS, where A is any vertex of the square and S is the center.

The inscribed circle is a circle that touches all sides of a square. It has a radius of a/2, where a is the length of the square's side.

Metr čtvereční

The square is also related to the basic unit of area, which is the square meter. What does this mean? If we say something has an area of one square meter, it means that the area is equivalent to that of a square with each side measuring one meter. If you have a room with an area of 10 square meters, it means that ten of these one-meter-side squares can fit there. Note, this does not mean the area of a square with side lengths of 10 meters! The following image illustrates this clearly:

If you took a square with a side length of 10 meters, it would represent one are. If the square had a side length of 100 meters, it would be a hectare.