Instruction manual
Determine the length of the rectangle. To do this, divide the given in the condition square to the width.
Perimeter calculate the rectangle using the formula P \u003d 2L + 2S, where P is the desired perimeter; S - given in the condition width; L is the length calculated in clause 2.
A special case of a rectangle is a square. All four sides of the square are equal. Therefore, to calculate the perimeter, it is enough to know the size of one side. Calculate the perimeter of the square according to the formula P \u003d 4S, where P is the desired perimeter; S - given in the condition width.
A parallelogram is also a regular polygon. The sides in it are pairwise equal and parallel. It is impossible to calculate the side size of a parallelogram from a known area and the other side cannot. You need to know the angle between the sides of the parallelogram. The specified conditions are not enough to calculate the perimeter of the parallelogram.
Draw an arbitrary parallelogram. On the side with the known conditional size, lower the height from the top of the parallelogram. For given width and area, the height of the parallelogram is constant and equal to the quotient of dividing the area by width. The angle between the sides of the parallelogram is not specified by condition. As the angle changes, the size of the unknown side of the parallelogram will change. Thus, the problem has many solutions.
A perimeter is the sum of the lengths of the sides of a geometric shape. In other words, if you take a thread and put it on the table, for example, a square, and then measure the length of this thread, then the resulting figure will be the perimeter of this square. Everyone knows what a perimeter is, but not everyone can immediately figure out how to calculate it.
There are various methods for measuring the perimeter of different figures.
Instruction manual
Square. It is well known that a square has 4 sides and they are equal. Therefore, the formula for calculating its perimeter looks like this:
where a is the length of one side of the given figure.
Simply put, measure one of the sides of the square and multiply this figure by the number of sides, that is, 4. In our case, the perimeter is 16 cm (4 * 4).
Rectangle and rhombus. For these two figures, only the sides parallel to each other are equal, respectively, the perimeter is determined as follows:
where a and b are the touching sides. Thus, in our example, the perimeter of the rectangle is 24 cm (2 * (8 + 4)).
Triangle. Since triangles are completely different - isosceles, irregular, with right angles, the only true way to determine the perimeter of such a figure is the formula:
That is, to calculate the perimeter of a triangle, simply measure the lengths of all three sides and add the resulting numbers. In our case, the perimeter of the triangle is 10.7 cm (2 + 5 + 3.7).
A circle . The perimeter of a circle is called the circumference, which is calculated by a special formula:
where d is the diameter of the circle, and 3.14 is the number "pi", which is specially derived by scientists to determine the perimeter of this geometric figure. Our circle (see the figure) has a diameter of 3 cm, that is, the circumference of the circle is 9.42 cm (3 * 3.14).
Sources:
- how to find the circumference
One of the values \u200b\u200bof the polygon is its perimeter. It is known from the school course of geometry that perimeter any polygon equal to the sum the lengths of all its sides. A rectangle is a kind of polygon, so the task of finding it perimeterit boils down to several actions.
Instruction manual
Sources:
- how to calculate the sum of the perimeter
Perimeter (P) is the sum of the lengths of all sides of the figure, and the quadrangle has four. So, to find the perimeter of a quadrangle, you just need to add the lengths of all its sides. But such figures as a rectangle, square, rhombus, that is, regular quadrangles, are known. Their perimeters are determined in special ways.
Instruction manual
If this figure is a rectangle (or parallelogram) of the AWSD, then it has the following properties: parallel sides are equal in pairs (see figure). AB \u003d SD and AC \u003d VD. Knowing this aspect ratio in this figure, we can derive the perimeter rectangle (and parallelogram): P \u003d AB + SD + AC + VD. Let some sides be equal to the number a, others to the number b, then P \u003d a + a + b + b \u003d 2 * a \u003d 2 * b \u003d 2 * (a + b). Example 1. In the rectangle AVSD, the sides are equal to AB \u003d SD \u003d 7 cm and AC \u003d VD \u003d 3 cm. Find the perimeter of such a rectangle. Solution: P \u003d 2 * (a + c). P \u003d 2 * (7 +3) \u003d 20 cm.
When solving problems on the sum of the lengths of the sides with a figure called a square or rhombus, a slightly modified perimeter formula should be used. A square and a rhombus are figures having the same four sides. Based on the definition of the perimeter, P \u003d AB + SD + AC + VD and assuming the designation of the length by the letter a, then P \u003d a + a + a + a \u003d 4 * a. Example 2. The rhombus has a side length of 2 cm. Find its perimeter. Solution: 4 * 2 cm \u003d 8 cm.
If this quadrangle is a trapezoid, then in this case you just need to add the lengths of its four sides. P \u003d AB + SD + AC + VD. Example 3. Find the perimeter of the trapezoid AVSD, if its sides are equal: AB \u003d 1 cm, SD \u003d 3 cm, AC \u003d 4 cm, VD \u003d 2 cm. Solution: P \u003d AB + SD + AC + VD \u003d 1 cm + 3 cm + 4 cm + 2 cm \u003d 10 cm. It may happen that the trapezoid is equilateral (it has two equal sides), then its perimeter can be reduced to the formula: P \u003d AB + SD + AC + VD \u003d a + b + a + c \u003d 2 * a + b + s. Example 4. Find the perimeter of an isosceles trapezoid if its side faces are 4 cm and the bases 2 cm and 6 cm. Solution: P \u003d 2 * a + b + c \u003d 2 * 4 cm + 2 cm + 6 cm \u003d 16 cm.
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Helpful advice
No one bothers to find the perimeter of a quadrangle (and any other figure), as the sum of the lengths of the sides, without using the derived formulas. They are given for convenience and simplification of the calculation. The solution method is not a mistake; the correct answer and knowledge of mathematical terminology are important.
Sources:
- how to find the perimeter of a rectangle
Although the word "perimeter" comes from the Greek designation of a circle, it is customary to name the total length of the borders of any flat geometric figure, including a square. The calculation of this parameter, as a rule, is not difficult and can be carried out in several ways, depending on the known source data.
Instruction manual
If the length of the side of the square (t) is known, then to find its perimeter (p), simply increase this value four times: p \u003d 4 * t.
If the side length is unknown, but the diagonal length (c) is given under the conditions of the problem, then this is sufficient to calculate the length of the sides, and therefore the perimeter (p) of the polygon. Use the Pythagorean theorem, which states that the square of the length of the long side of a right triangle (hypotenuse) is equal to the sum of the squares of the length of the short sides (legs). In a right-angled triangle made up of two adjacent sides of a square and connecting the extreme points of a segment, the hypotenuse coincides with the diagonal of the quadrangle. It follows that the length of the side of the square is equal to the ratio of the length of the diagonal to the square root of two. Use this expression in the formula to calculate the perimeter from the previous step: p \u003d 4 * c / √2.
If only the area (S) of the square of the plane segment bounded by the perimeter is given, then this will be enough to determine the length of one side. Since the area of \u200b\u200bany rectangle is equal to the product of the lengths of its adjacent sides, then to find the perimeter (p), take the square root of the area, and increase the result four times: p \u003d 4 * √S.
If the radius of the circle (R) described near the square is known, then to find the perimeter of the polygon (p), multiply it by eight and divide the result by the square root of two: p \u003d 8 * R / √2.
If the circle whose radius is known is inscribed in a square, then calculate its perimeter (p) by simply multiplying the radius (r) by eight: P \u003d 8 * r.
If the square under consideration in the conditions of the problem is described by the coordinates of its vertices, then to calculate the perimeter you will need data on only two vertices belonging to one of the sides of the figure. Determine the length of this side, proceeding from the same Pythagorean theorem for a triangle composed of itself and its projections on the coordinate axis, and increase the result by four times. Since the lengths of the projections onto the coordinate axes are equal to the absolute value of the differences of the corresponding coordinates of the two points (X₁; Y₁ and X₂; Y₂), the formula can be written as follows: p \u003d 4 * √ ((X₁-X₂) ² + (Y₁-Y₂) ²) .
Any convex and flat geometric figure has a line bordering its internal space - the perimeter. For polygons, it consists of individual segments (sides), the sum of the lengths of which determines the length of the perimeter. The portion of the plane bounded by this perimeter can also be expressed in terms of the lengths of the sides and the angles at the vertices of the figure. Below are the corresponding formulas for one of the types of polygons - a parallelogram.
Instruction manual
If, under the conditions of the problem, the lengths of two adjacent sides of the parallelogram (a and b) and the angle between them (γ) are given, then this will be sufficient to calculate both parameters. To calculate the perimeter (P) of the quadrangle, add the lengths of the sides and double the resulting value: P \u003d 2 * (a + b). It is necessary to calculate the area (S) of the figure using the trigonometric function - the sine. Multiply the lengths of the sides, and multiply the result by the sine of the known angle: S \u003d a * b * sin (γ).
If the length of only one side (a) of the parallelogram is known, but there is data on the height (h) and the magnitude of the angle (α) at \u200b\u200bany of the vertices of the polygon, this will allow us to find both the perimeter (P) and area (S). The sum of all the angles in any quadrangle is 360 °, and in a parallelogram, those that lie at opposite vertices are the same. Therefore, to find the value of the angle remaining unknown, subtract the value of the known from 180 °. After that, consider a triangle made up of a height and an angle lying opposite it, the values \u200b\u200bof which are known, as well as the sides that are still unknown. Apply the sine theorem to it, and find out that the length of the side will be equal to the ratio of the height to the sine of the angle opposite it: h / sin (α).
After preliminary calculations of the previous step, make the necessary formulas. Substitute the resulting expression in the formula for finding the perimeter from the first step and get the following equality: P \u003d 2 * (a + h / sin (α)). If the height connects two opposite sides of the parallelogram, the length of which is given in the initial conditions, to find the area, simply multiply these two values: S \u003d a * h. If this condition is not met, then substitute in the formula the expression for the other side obtained in the previous step: S \u003d a * h / sin (α).
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The perimeter of a polygon is the sum of all its sides. Accordingly, to find this value, you need to add all sides of the polygon. For some types of polygon, there are special formulas that allow you to do this faster.
You will need
- - ruler;
- - Pythagorean theorem;
- - calculator.
Instruction manual
Using a ruler, or in any other way, measure the lengths of all sides polygon. Then add the values \u200b\u200bobtained as a result of the measurements and get the perimeter of this geometric figure. For example, if the sides of a triangle are 12, 16, and 10 cm, then its perimeter will be 12 + 16 + 10 \u003d 38 cm.
In general, the perimeter of any correct polygon (this is convex polygonwhose sides are equal to each other) is equal to the length of one side multiplied by the number of its sides or angles (this number is equal to each other for all polygons, for example, an octagon has 8 angles and 8 sides). For example, to find the perimeter of a regular hexagon with a side of 3 cm, multiply it by 6 (P \u003d 3 ∙ 6 \u003d 18 cm).
To find the perimeter of a rectangle or parallelogram whose opposite sides are parallel and equal, measure the lengths of their unequal sides a and b. In the case of a rectangle, this is its length and width. Then find their sum, and multiply the resulting number by 2 (P \u003d (a + b) ∙ 2). For example, if there is a rectangle with sides 4 and 6 cm, which are its length and width, find its perimeter by the formula P \u003d (4 + 6) ∙ 2 \u003d 20 cm.
If only two sides are given in a right triangle, find the third using the Pythagorean theorem. After that, find the sum of all sides - this will be its perimeter. For example, if the legs of a right-angled triangle are a \u003d 6 cm and b \u003d 8 cm, find the sum of their squares, and extract the square root from the result. This will be the length of the third side (hypotenuse), c \u003d √ (6² + 8²) \u003d √ (36 + 64) \u003d √100 \u003d 10 cm. Calculate the perimeter as the sum of the three sides of the triangle P \u003d 6 + 8 + 10 \u003d 24 cm.
About the square rectangle begin to speak in elementary grades. There are various formulas with which you can calculate it. Let's consider some of them.
Square (from lat. quadratus - quadrangular) - a true quadrangle in which all sides and angles are equal to each other. It can be defined as a rectangle in which two adjacent sides are equal to each other, or as a rhombus in which all angles are straight.
Symmetry.A square has greater symmetry in the middle of all quadrangles. He has:
In that case, indicate the side of the square athen the diagonal length d calculated by the Pythagorean axiom:
d \u003d √ (a2 + a2) \u003d √ (2a2) \u003d √2
Inscribed and circumscribed circles.The circle inscribed in the square touches the middle of all sides of the square and has a radius requal to half the side of the square a.The circle circumscribed around the square passes through all its vertices and has a radius Requal to half the length of the diagonal of the square d:
r \u003d a / 2,
R \u003d d / 2 \u003d (√2 / 2)
Perimeter and area.Perimeter P square consists of the lengths of 4 of its sides. Square S a square is the square of the length of its side:
P \u003d 4a \u003d 8r \u003d 2√2 · R,
S \u003d a2 \u003d 4r2 \u003d 2R2.
Sources:
Material Source Website
The ability to find the perimeter is required not only for students to solve mathematical problems, but also, for example, during repairs. The perimeter is the sum of the lengths of the sides of the figure. If the room is a square, then it is logical to assume that each side of the square will be equal in length. In order to revive the knowledge of geometry, recall the definition of a square. This is a geometric figure, all sides of which are equal and are in relation to each other at an angle of 90 degrees.
There is a standard formula for calculating the perimeter. It is necessary to add the length of all sides of the figure. And it doesn’t matter what shape it is - a trapezoid, a rectangle or a square. That is, if the length of one side of the square is 5 meters, then we add four times five times and get 20 meters. Of course, it’s easier and more correct not to add but to multiply. Multiply the side length of the square, 5 meters in this case, by the number of sides. We get P \u003d 4a. Where a is the length of one of the sides. In our case, it is 5 * 4 \u003d 20.
There is another option for finding the perimeter, if you do not want or do not have the opportunity to measure the length of the side, but you know the area. For example, the area of \u200b\u200byour square room is 36 square meters. First we need to find out the length of one side. To do this, remember how the square area is calculated. The square area is calculated by squaring the length of its side into a square (S \u003d a2). That is, if the length of the side of the square is 8, then its area will be 8 * 8 \u003d 64.
It is logical that to find out the length of one of the sides we will need to extract the square root of the area. a \u003d √36. The square root of 36 is 6. To test 6, we squash it and get 36, everything is correct. Now, knowing the length of the side of the square, we can find its perimeter. For this, we also multiply the length of the side by the number of sides of the square 4. We get P \u003d 4 * 6 \u003d 24.
Thus, if you want to lay skirting boards around the entire perimeter of your room, you can easily calculate how much material you need to purchase.
Another method is often used in mathematics. If you know the length of the diagonal of the square, then you can also find the length of its side, and then calculate the perimeter. The calculations are based on the Pythagorean theorem. Since a square when dividing it diagonally forms two isosceles triangles, we conclude from the theorem that the square of the diagonal length of a square is equal to two squares of its side. Thus, if the length of the diagonal is 10 cm, then we obtain the following equation. 10 squared, we get 100. 100 is the square of the diagonal. The length of the side in this case will be 2a2. We get that 100 \u003d 2a2. Next, we solve the equation: 50 \u003d a2. a \u003d square root of 50. This is about 7. The length of the side is 7, which means the perimeter will be 7 * 4 \u003d 28. This example is used less often, but can sometimes be useful.
| 13.11.2014 |
A square is a regular quadrangle (or rhombus) in which all angles are straight and the sides are equal to each other. Like any other regular polygonat square can be calculated perimeter and square. If a square square already known then find its sides and then and perimeter not be difficult.
Instruction manual
Square square found by the formula:
S \u003d a?
This means that in order to calculate square square, you need to multiply the lengths of its two sides by each other. As a result, if you know square square, then when extracting the root from this value, you can find out the length of the side square.
Example: square square 36 cm? To find out the side of the given square, it is necessary to extract the square root of the area value. So the side length of a given square 6 cm
To find perimeterbut square it is necessary to add the lengths of all its sides. Using a formula, this can be expressed as follows:
P \u003d a + a + a + a.
If you extract the root from the area squareand then add the resulting value 4 times, then you can find perimeter square.
Example: Dan square with squareu 49 cm ?. Wanted to find him perimeter.
Decision:
First you need to extract the square root square:? 49 \u003d 7 cm
Then, calculating the length of the side squarecan be calculated and perimeter: 7 + 7 + 7 + 7 \u003d 28 cm
Answer: perimeter square squareu 49 cm? is 28 cm
note
For a square, the following definitions are valid:
A square is a rectangle that has equal sides.
A square is a special kind of rhombus, in which each of the angles is 90 degrees.
Being a regular quadrangle, a circle can be described or inscribed around a square. The radius of a circle inscribed in a square can be found by the formula:
R \u003d t / 2, where t is the side of the square.
If the circle is described around it, then its radius is as follows:
R \u003d (? 2 * t) / 2
Based on these formulas, we can derive new ones to find the perimeter of the square:
P \u003d 8 * R, where R is the radius of the inscribed circle;
P \u003d 4 *? 2 * R, where R is the radius of the circumscribed circle.
The square is a unique geometric figure, because it is absolutely symmetrical, no matter how and where to draw the axis of symmetry.