Geometric figures. Square. What is a square? Square property definition signs
A square is a geometric figure with equal sides and angles. Most of us have known about this since school. But, unfortunately, not everyone remembers what properties it has and how its area and perimeter are calculated.
Therefore, in this article we will talk about what a square is in more detail.
Basic definition and properties of a square
So, a square is a regular quadrilateral (rectangle) with equal sides and angles. A rectangle is a parallelogram, therefore a square should also be considered a parallelogram. In addition, given that all sides of a square are the same length, it is also a rhombus. Thus, we can conclude that a square has some properties of both a rhombus and a rectangle.
What are the properties of a square? Firstly, all its angles are right, and the diagonals and sides of such a rectangle are equal to each other. Secondly, the diagonals of the square are not only mutually perpendicular, but also act as bisectors of the corners of the quadrilateral. At the point of intersection they are divided in half.
How to calculate the perimeter and area of a square
To calculate the area and perimeter of a square, you need to know the value of one side of a given rectangle or diagonal. Since its sides have the same length, in order to find out the perimeter of a square, you should multiply the side value by 4 or simply add all 4 sides: the resulting sum is the perimeter. For example, the length of one side of your square is 5 cm. Therefore, 5 needs to be multiplied by 4 (5 x 4 = 20) or added all sides: 5+5+5+5 = 20. This is the simplest way to calculate.
The perimeter of the square is also calculated using the diagonal value. First read our article on the topic. The perimeter of the square is equal to the product of the length of the diagonal by 2 roots of 2. This means that if the length of the diagonal of your square is 10 cm, then you should take the root from 2 (which will be approximately 1.4) and multiply by 2, then by the length. Thus, 1.4 x 2 x 10 = 28 cm (if rounded). That is, the perimeter of a square with a diagonal of 10 cm will be about 28 cm.
To calculate the area of a square, a simple method is used: square the length of one side. So, if it is 4 cm, then 4 should be multiplied by 4. It turns out that the area of a square with a side of 4 cm is equal to 16 cm.
- (Latin quadratum, from quadrare to make quadrangular). 1) rectangular, equilateral quadrangle. 2) a number that, when multiplied by itself, gives given number. 3) unit for measuring planes; eg: square feet, inches and... Dictionary foreign words Russian language
Squared. Jarg. they say Neglect About extremely stupid, hopeless stupid man. /i> Square is a stupid, slow-witted fellow. Nikitina 1996, 82. Square your hypotenuse! Jarg. school Bran. Expression of annoyance, irritation, indignation. VMN 2003, 62.… … Big dictionary Russian sayings
SQUARE, in biology, a square frame used to mark an area of surface for the purpose of studying the plants located on it. This area of soil itself is also called a square. As a rule, such a square is 0.5 or 1 m2. Using this... ... Scientific and technical encyclopedic dictionary
SQUARE, square, man. (lat. quadratus quadrangular). 1. Equilateral rectangle (mat.). 2. The shape of such a rectangle for some object (book). A brightly lit square window. 3. A quadrangular hart block is a measure for... ... Dictionary Ushakova
Husband. equilateral and rectangular quadrilateral; people call it a round quadrangle or a cage. Divide the area into squares, into sections of this type. | The square of a number, its product multiplied by itself. Square pattern or... ... Dahl's Explanatory Dictionary
In printing, 1) a unit of length used to measure fonts and typesetting format. 1 square = 48 points (approx. 18.05 mm). 2) A type of whitespace material for filling large gaps in lines ... Big Encyclopedic Dictionary
Parallelogram, cell, material, rectangle, degree, square Dictionary of Russian synonyms. square noun, number of synonyms: 9 hypercube (12) ... Synonym dictionary
square- SQUARE, a, m. Prison; camera. trample square to be in prison, cell. From ug... Dictionary of Russian argot
square- (Quad) 1. One of the basic units of Didot’s typometric system, equal to 4 ciceros, or 48 points. 1 square is equal to 18.048 mm. 2. Space material used in the manufacture of typesetting printing plates using letterpress printing. Squares are distinguished by... Font terminology
"Square"- “Kvadrat”, a club for jazz music lovers (jazz club). Created in 1964 at the Lensovet House of Culture (since 1965 it was housed in the S. M. Kirov House of Culture, and since 1986 at the Youth Palace). Unites musicians and lovers of classical jazz. "Square" continued... ... Encyclopedic reference book "St. Petersburg"
- (from the Latin quadratus quadrangular), 1) an equilateral rectangle. 2) The second power a2 of the number a (the name is due to the fact that this is how the area of a square with side a is expressed) ... Modern encyclopedia
Books
- Square. From the history of Russian jazz. The book contains selected materials on the history of Russian jazz, published in the 60-80s of the last century on the pages of the legendary unofficial samizdat typewritten magazine...
- Square, Willi Karlsson. The book by a prominent figure of the Communist Party of Denmark can be called a true chronicle of the labor movement in the country during a turbulent era from the beginning of the crisis of the 1930s until the occupation of Denmark by the Nazis.…
Square is a quadrilateral with equal sides and angles.
Diagonal of a square is a segment connecting its two opposite vertices.
A parallelogram, rhombus and rectangle are also a square if they have right angles, equal lengths of sides and diagonals.
Properties of a square
1. The lengths of the sides of the square are equal.
AB=BC=CD=DA
2. All angles of the square are right.
\angle ABC = \angle BCD = \angle CDA = \angle DAB = 90^(\circ)
3. Opposite sides of the square are parallel to each other.
AB\parallel CD, BC\parallel AD
4. The sum of all angles of a square is 360 degrees.
\angle ABC + \angle BCD + \angle CDA + \angle DAB = 360^(\circ)
5. The angle between the diagonal and the side is 45 degrees.
\angle BAC = \angle BCA = \angle CAD = \angle ACD = 45^(\circ)
Proof
The square is a rhombus \Rightarrow AC is the bisector of angle A and it is equal to 45^(\circ) . Then AC divides \angle A and \angle C into 2 angles of 45^(\circ) .
6. The diagonals of a square are identical, perpendicular and bisected by the point of intersection.
AO = BO = CO = DO
\angle AOB = \angle BOC = \angle COD = \angle AOD = 90^(\circ)
AC = BD
Proof
Since a square is a rectangle \Rightarrow the diagonals are equal; since - rhombus \Rightarrow diagonals are perpendicular. And since it is a parallelogram, \Rightarrow diagonals are divided in half by the intersection point.
7. Each of the diagonals divides the square into two isosceles right triangles.
\triangle ABD = \triangle CBD = \triangle ABC = \triangle ACD
8. Both diagonals divide the square into 4 isosceles right triangles.
\triangle AOB = \triangle BOC = \triangle COD = \triangle AOD
9. If the side of the square is equal to a, then the diagonal will be equal to a \sqrt(2) .
When they have the same lengths of diagonals, sides and equal angles.
Properties of a square.
All 4 sides of the square have the same length, i.e. the sides of the square are equal:
AB = BC = CD = AD
Opposite sides of the square are parallel:
AB|| CD, B.C.|| AD
All diagonals divide the corner of the square into two equal parts, thus they turn out to be bisectors of the corners of the square:
ΔABC = ΔADC = ΔBAD = ΔBCD
∠ ACB =∠ ACD =∠ BDC =∠ BDA =∠ CAB =∠ CAD =∠ DBC =∠ DBA = 45°
The diagonals divide the square into 4 identical triangles, in addition, the resulting triangles are both isosceles and right-angled:
ΔAOB = ΔBOC = ΔCOD = ΔDOA
Diagonal of a square.
Diagonal of a square is any segment that connects the 2 vertices of opposite corners of a square.
The diagonal of any square is √2 times greater than the side of this square.
Formulas for determining the length of the diagonal of a square:
1. Formula for the diagonal of a square in terms of the side of the square:
2. Formula for the diagonal of a square in terms of the area of the square:
3. Formula for the diagonal of a square through the perimeter of a square:
4. Sum of square angles = 360°:
5. Diagonals of a square of the same length:
6. All diagonals of a square divide the square into 2 identical figures that are symmetrical:
7. The angle of intersection of the diagonals of a square is 90°, intersecting each other, the diagonals are divided into two equal parts:
8. Formula for the diagonal of a square using the length of a segment l:
9. Formula for the diagonal of a square in terms of the radius of the inscribed circle:
R- radius of the inscribed circle;
D- diameter of the inscribed circle;
d- diagonal of a square.
10. Formula for the diagonal of a square in terms of the radius of the circumscribed circle:
R- radius of the circumscribed circle;
D- diameter of the circumscribed circle;
d- diagonal.
11. Formula for the diagonal of a square through a line that extends from the corner to the middle of the side of the square:
C- a line that extends from the corner to the middle of the side of the square;
d- diagonal.
Inscribed circle in a square- this is a circle adjacent to the midpoints of the sides of the square and having a center at the intersection of the diagonals of the square.
Inscribed circle radius- side of the square (half).
Area of a circle inscribed in a square less than the area of the square by π/4 times.
Circle circumscribed around a square- this is a circle that passes through the 4 vertices of the square and which has a center at the intersection of the diagonals of the square.
Radius of a circle circumscribed around square greater than the radius of the inscribed circle by √2 times.
Radius of a circle circumscribed around a square equal to 1/2 diagonal.
Area of a circle circumscribed around a square the larger area of the same square is π/2 times.