Often on Internet you can find taunts about how knowledge in mathematics – integrals, differentials, trigonometric functions and other topics of the subject does not help relieve a person's life. Such jokes are in vain, because it helps knowing how to calculate the perimeter of a square, rectangle, and other geometrical figures in construction work. Material: tiles, Wallpaper, floor coverings not to determine without an understanding of basic mathematical formulas and geometric shapes.
Properties of a square
Any calculation in mathematics are based on the properties of the object. To answer the question: "What is the perimeter of the square?" – it is recommended to remember the distinctive characteristics of this shape.
- The equality of the parties.
- Four corners value of 90 degrees.
- The parallelism of the sides.
- The rotary symmetry. When you rotate the figure view is unchanged.
- The ability to describe and inscribe a circle.
- Diagonal crossing divide each other in half.
- The area of the figure characterizes filled a square place in two-dimensional space.
- The perimeter of a figure is nothing but the sum of the lengths of its sides.
- From the previous property it follows that the units of measurement of the perimeter will be units of length: m, cm, dm, and others.
To calculate skirting boards to complete the repair in a square room, you need to know the length of the room. It is necessary to calculate its perimeter.
In Greek language the word means "measure around". The term applies to all closed shapes: square, circle, rectangle, triangle, trapezoid, and other. Knowledge, by definition, perimeter of basic figures necessary for solving complex geometric problems with objects of irregular shape. For example, to calculate the baseboards in the room layout type "G", or as they call it, "boot", you will need to determine the perimeter of square and rectangle. Because the shape of the room is composed of these basic shapes.
Common designation of such magnitude – the letter R. Each figure based on its properties has a different formula for determining the perimeter.
- Equality of opposite sides.
- The equality of the diagonals.
- The ability to describe a circle.
- The height of the rectangle is equal to its sides.
- The sum of the angles is equal to 360 degrees, and all corners are straight.
- The parallelism of the opposite sides.
- The perpendicular adjacent sides.
- The sum of the squares of the diagonals of the rectangle is equal to the sum of the squares of its sides.
- The intersecting diagonals divide each other in half.
- The inability to fit into the shape of a circle.
The perimeter of the square
Depending on the (known) parameters of the square, there are different formulas for determining the perimeter. A simple task is to calculate the perimeter when the installed length of its side (s). In this case, P=s s s s or 4*C. for Example, the length of the square side 7 cm, then the perimeter of a figure bude 28 cm (4*7).
In the first case everything is clear, but how to find the perimeter of a square, knowing its area? And everything is very clear. Since the area of the figure is determined by multiplying one side to the other and in a square all sides are equal, you must remove the root from the known values. Example: there is a square with area of 25 DM 2. Root of 25 is 5 – this value describes the length of a side of a square. Now, substituting the found value is 5 DM 2 in the original formula for the perimeter, we can solve the problem. The answer is the value 20. That is 4 multiplied by 5, get the required amount.
A square and a circle
Of the properties under consideration of the figures suggests that a square can be inscribed circle, and also to describe it around the figure.
A feature of the inscribed circle is the division of sides of a square in half. Therefore, the radius equals half the length of the sides of the square. Then the side=2*radius. The perimeter of the square in this case is equal to 4*2*radius or 8 times the radius of the circle.
The perimeter of the rectangle
The most elementary formula for determining the perimeter of a rectangle using the known values of sides is: P=2(a b), where a and b are the lengths of the sides of the figure.
The diagonal of the rectangle similar to a square divides the figure in half, forming a rectangular triangle. However, the task is complicated by the fact that the sides of the triangle are unequal. In the case of the known value of one of the sides and diagonal, the second can be found by following the Pythagorean theorem: d 2 =a 2 in 2. where a and b – side of the figure, and d is diagonal.
If neither one of the parties, then in the case enters a knowledge of trigonometry: sines, cosines and other functions.
Finding the perimeter of the circumscribed circle and the known diameter is that diameter equal to the length of the diagonal of the figure. Then the solution of the problem defined by the presence of known quantities. If the given angles, then using trigonometric functions. If given a side, the answer will be found using the Pythagorean theorem.
The rectangle and the trigonometric functions
For clarity, an example is given of solving the problem. Given: rectangle AVSDd ) 20 cmf – 30°. To find the perimeter of shapes.
To calculate the perimeter you must find the second side of the shape. Through the Pythagorean theorem, as known length of the hypotenuse and of one of the legs or, again, using the aspect ratio for the cosine of an angle.
The cosine of the angle f is expressed as the ratio of the adjacent sides to the hypotenuse and is equal to √3/2.
√3/2=n/d. n=(d *√3)/2 or 10*√3. After extraction of the square root of 3, we get the length of a side of triangle: 10*1,73=17,3 cm
The perimeter is 2(17,3 10)=2*27,3=54,6 see
Perimeter and Attitude
In the curriculum there are challenges in geometry, when the lengths of the sides of the rectangle expressed their attitude to each other. Consideration of solving such problems are presented below.
It is known that the sum of the lengths of all the sides of the rectangle, that is, its perimeter is 84 cm length (l) to width (W) of 3:2. To find the figure.
Solution: let the length be 3x and the width is 2, according to the ratio given in the problem statement. The formula for the perimeter of a rectangle with the obtained data of the lengths of the sides will be the following: 3x 3x 2x 2x = 84. Then, 10x = 84, x=8,4 cm x Substituting into the expression for the length and width of a rectangle, you can find the desired value. Length will be: 3*8,4 = 25.2 cm
The article is devoted to solving the most common tasks in the school curriculum. And it's not all ways of finding the perimeter of square and rectangle.