The rule is what the perimeter is. Simple task: how to find the perimeter

Surely each of us learned at school such an important component of geometry as perimeter. Finding the perimeter is simply necessary for solving many problems. Our article will tell you how to find the perimeter.

It is worth remembering that the perimeter of any figure is almost always the sum of its sides. Let's look at a few different geometric shapes.

1. A rectangle is a quadrilateral whose parallel sides are equal in pairs. If one side is X and the other is Y, then we get the following formula for finding the perimeter of this figure:

   P = 2(X+Y) = X+Y+X+Y = 2X+2Y.

   An example of solving a problem:

   Let's assume that side X = 5 cm, side Y = 10 cm. So, substituting these values ​​into our formula, we get - P = 2*5 cm + 2* 10cm = 30 cm.

2. A trapezoid is a quadrilateral whose two opposite sides are parallel but not equal to each other. The perimeter of a trapezoid is the sum of all four sides:

   P = X+Y+Z+W, where X, Y, Z, W are the sides of the figure.

   An example of solving a problem:

   Let's assume that side X = 5 cm, side Y = 10 cm, side Z = 8 cm, side W = 20 cm. So, substituting these values ​​into our formula, we get - P = 5 cm + 10 cm + 8 cm + 20 cm = 43 cm.

3. The perimeter of a circle (circumference) can be calculated using the formula:

   P = 2rπ = dπ, where r is the radius of the circle, d is the diameter of the circle.

   An example of solving a problem:

   Let's assume that the radius r of our circle is 5 cm, then the diameter d will be equal to 2 * 5 cm = 10 cm. It is known that π = 3.14. This means that by substituting these values ​​into our formula, we get - P = 2*5 cm*3.14 = 31.4 cm.

4. If you need to find the perimeter of a triangle, then you may encounter a number of problems in doing so, since triangles can have very different shapes. For example, there are acute, obtuse, isosceles, right and equilateral triangles. Although the formula for all types of triangles is:

   P = X+Y+Z, where X, Y, Z are the sides of the figure.

   The problem is that when solving many problems to find the perimeter of this figure, you will not always know the lengths of all sides. For example, instead of information about the length of one of the sides, you can have the degree of an angle or the length of the height of a particular triangle. This will significantly complicate the task, but will not make its solution unrealistic. You can read “” about how to find the perimeter of a triangle, no matter what shape it is.

5. The perimeter of a figure such as a rhombus is found in the same way as the perimeter of a square, because a rhombus is a parallelogram that has equal sides. You can find out how to find the perimeter of a square by reading the article on our website "".

   Now you know how to find the side of the perimeter of the geometric figure you need!

There are several concepts of perimeter.

Geometric: every closed plane has the length of its boundaries. And from the security area. That is, the perimeter is the actual protected border or territory of the protected object. Since this topic is from the Education section and not from the Law and Safety section, we should focus on the geometric concept of the perimeter.

So what is perimeter?

For some reason this question baffles some young people. Didn't they learn this in school? If some mathematical (geometric) formulas that are fed to schoolchildren will never be useful in life, then knowing what a perimeter is is simply necessary, and this knowledge, you can be sure, will be in demand.

What is the perimeter of your country house? What about the plot? The area of ​​both depends on the perimeter. What if your vegetable garden, field, or garden has an oval shape or many corners? How do you find out their perimeter?

First you should look at dictionaries and encyclopedias. And understand for yourself what the concept of “perimeter” includes.

The Large Encyclopedic Dictionary gives the following definition of perimeter: this is the length of a contour that is closed. The sum of the lengths of the sides of a geometric figure, for example, all five sides of a pentagon.

Let's say there is a plot of land that is a pentagon. One side extends 20 meters, the other 16 meters, the third 4 meters, the fourth 11 meters and the fifth 6 meters. What is the perimeter of the land plot? By simple arithmetic addition we calculate the perimeter of the land plot: 20 + 16 + 4 + 11 + 6 = 57 meters.

Ushakov's dictionary gives the following explanation for the concept of “perimeter”: it is the sum of the lengths of all sides of a flat figure. Which we have already illustrated in the above example.

What about the circle? It's also flat. What is its perimeter, and how to calculate it?

There is a formula for calculating the perimeter (length) of a circle. But to do this, you first need to remember what a circle is and what elements it has. And a circle is a curve that is not only flat and closed, but also all its points are located at the same distance from a given point, called the center.

The straight line segment connecting this center with any point on the circle is the radius (R).

A straight line segment passing through the center of a circle and connecting its two points furthest from each other is the diameter (D). The diameter is equal to two radii.

The ratio of the circumference of a circle to its diameter is the same for any circle and is equal to the constant number 3, 14... This number is denoted by the letter π (pi).

Now we can give a formula for calculating the perimeter (length) of a circle: P = 2πR or π D.

Let's say we know the radius of the circle: 5 meters. What will its perimeter be equal to?

The actions here will be as follows: multiply the diameter (10 meters) by 3.14. And we get the perimeter of the circle equal to 31.4 meters.

There are also more complex figures, the perimeter of which needs to be known. Here, mathematical analysis methods are used to calculate the perimeter, which requires special knowledge...

Instructions

Sources:

• how to find perimeter abcd

Perimeter is the sum of the lengths of the sides of a geometric figure. In other words, if you take a thread and use it to lay out, for example, a square on a table, 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 different ways to measure the perimeter of different shapes.

Instructions

Square. It is well known that a square has 4 sides and they are . Therefore, to calculate its perimeter like this:

where a is the length of one side of this figure.

Simply put, measure one of the sides of the square and multiply this figure by the number of sides, that is, by 4. In our case, it is 16 cm (4 * 4).

Rectangle and rhombus. For these two figures, only the sides parallel to each other are equal, so 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 can be completely different - isosceles, irregular, with angles, the only correct 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).

The perimeter is called the circumference, which is calculated using a special formula:

where d is a circle, and 3.14 is the number “pi”, which was specially derived by scientists to determine the perimeter of a given geometric figure. Our circle (cm.) has 3 cm, that is, the perimeter of the circle is 9.42 cm (3 * 3.14).

Sources:

• how to find the circumference of a circle

In general, ohm is the length of the line that limits a closed figure. For polygons, the perimeter is the sum of all side lengths. This value can be measured, and for many figures it can be simply calculated if the lengths of the corresponding elements are known.

You will need

• - ruler or tape measure;
• - strong thread;
• - roller rangefinder.

Instructions

To measure an arbitrary polygon, use a ruler or other measuring device to measure all its sides, and then find their sum. If given a quadrilateral with sides of 5, 3, 7 and 4 cm, which are measured with a ruler, find the perimeter by adding them together P = 5 + 3 + 7 + 4 = 19 cm.

If the figure is arbitrary and includes more than just straight lines, then measure its perimeter with a regular rope or thread. To do this, position it so that it exactly follows all the lines limiting the figure, and make a mark on it, if possible, just cut it off to avoid confusion. Then, using a tape measure or ruler, measure the length of the thread, it will be equal to the perimeter of this figure. Be sure to ensure that the thread follows the line as closely as possible for greater accuracy of the result.

Measure the perimeter of a complex geometric figure with a roller rangefinder (curvimeter). To do this, a point is marked on a line at which the rangefinder roller is installed and rolled along it until it returns to the starting point. The distance measured by the roller rangefinder will be equal to the perimeter of the figure.

Calculate the perimeter of some geometric shapes. For example, to find the perimeter of any regular polygon (a convex polygon whose sides), multiply the length of the side by the number of angles or sides (they are equal). To find the perimeter of a regular triangle with a side of 4 cm, multiply this by 3 (P=4∙3=12 cm).

To find the perimeter, add up the lengths of all its sides. If all the sides are not given, but their angles are, find them using the sine or cosine theorem. If two sides of a right triangle are known, find the third using the Pythagorean theorem and find their sum. For example, if it is known that the legs of a right triangle are 3 and 4 cm, then the hypotenuse will be equal to √(3²+4²)=5 cm. Then the perimeter P=3+4+5=12 cm.

Sources:

• perimeter

To solve this problem using vector algebra methods, you need to know the following concepts: geometric vector sum and scalar product of vectors, and you should also remember the property of the sum of interior angles of a quadrilateral.

You will need

• - paper;
• - pen;
• - ruler.

Instructions

A vector is a directed segment, that is, a quantity that is considered fully specified if its length and direction (angle) to a given axis are given. The position of the vector is no longer limited by anything. Two vectors with lengths and the same direction are considered equal. Therefore, when using coordinates, vectors are represented by radius vectors of the points of its end (the origin is at the origin of coordinates).

By definition: the resulting vector of a geometric sum of vectors is a vector that starts from the beginning of the first and has the end of the second, provided that the end of the first is combined with the beginning of the second. This can be continued further, building a chain of similarly located vectors.
Draw the given ABCD with vectors a, b, c and d in Fig. 1. Obviously, with this arrangement the resulting vector is d=a+ b+c.

In this case, the scalar product is more convenient based on the vectors a and d. Dot product, denoted by (a, d)= |a||d|cosф1. Here φ1 is the angle between vectors a and d.
The dot product of vectors given by coordinates is defined by the following:
(a(ax, ay), d(dx, dy))=axdx+aydy, |a|^2= ax^2+ ay^2, |d|^2= dx^2+ dy^2, then
cos Ф1=(axdx+aydy)/(sqrt(ax^2+ ay^2)sqrt(dx^2+ dy^2)).

The basic concepts of vector algebra in relation to the problem at hand lead to the fact that for an unambiguous formulation of this problem, it is sufficient to specify three vectors located, say, on AB, BC, and CD, that is, a, b, c. You can, of course, immediately set points A, B, C, D, but this method is redundant (4 parameters instead of 3).

Example. The quadrilateral ABCD is defined by the vectors of its sides AB, BC, CD a(1,0), b(1,1), c(-1,2). Find the angles between its sides.
Solution. In connection with the above, 4th vector (for AD)
d(dx,dy)=a+ b+c=(ax+bx +cx, ay+by+cy)=(1,3). Following the method of calculating the angle between vectors a
cosф1=(axdx+aydy)/(sqrt(ax^2+ ay^2)sqrt(dx^2+ dy^2))=1/sqrt(10), ф1=arcos(1/sqrt(10)).
-cosф2=(axbx+ayby)/(sqrt(ax^2+ ay^2)sqrt(bx^2+ by^2))=1/sqrt2, Ф2=arcos(-1/sqrt2), Ф2=3п/ 4.
-cosф3=(bxcx+bycy)/(sqrt(bx^2+ by^2)sqrt(cx^2+ cy^2))=1/(sqrt2sqrt5), Ф3=arcos(-1/sqrt(10)) =p-f1.
In accordance with Remark 2 - ph4=2p- ph1 - ph2- ph3=p/4.

Video on the topic

note

Remark 1. The definition of the scalar product uses the angle between the vectors. Here, for example, φ2 is the angle between AB and BC, and between a and b this angle is n-φ2. cos(n- ph2)=- cosph2. Likewise for f3.
Remark 2. It is known that the sum of the angles of a quadrilateral is 2n. Therefore, φ4 = 2n- φ1 - φ2- φ3.

Any convex and flat geometric figure has a line limiting 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 section of the plane limited 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.

Instructions

If the problem is given the lengths of two adjacent sides of a parallelogram (a and b) and the size of the angle between them (γ), then this will be enough to calculate both parameters. To calculate the perimeter (P) of a quadrilateral, add the lengths of the sides and double the resulting value: P = 2*(a+b). You will have to calculate (S) figures using the trigonometric function - sine. Multiply the lengths of the sides, and multiply the result by the known angle: S = a*b*sin(γ).

If the length of only one of the sides (a) of the parallelogram is known, but there is data on (h) and the magnitude of the angle (α) at ​​any of the vertices, then the perimeter (P) (S) will also allow this. The sum of all angles in any one is equal to 360°, and in a parallelogram those that lie at opposite vertices are the same. Therefore, to find the value of the remaining unknown angle, subtract the known value from 180°. After this, consider a triangle made up of an altitude and an angle opposite it, the values ​​of which are known, as well as a side that is 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 carrying out the preliminary calculations of the previous step, make the necessary ones. Substitute the resulting expression into the formula from the first step and get the equality: P = 2*(a+h/sin(α)). In the event that 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=a*h. If this condition is not met, then substitute into the formula the expression for the other side obtained in the previous step: S=a*h/sin(α).

Video on the topic

Among the main tasks of analytical geometry, the first place is represented by geometric inequalities, equations or systems of one or another. This is possible through the use of coordinates. An experienced mathematician, just looking at the equation, can easily tell which geometric figure can be drawn.

Instructions

The equation F (x, y) can be used to define a curve or a straight line if two conditions are met: if the coordinates of a point that does not belong to the given line do not satisfy the equation; if each point of the desired line with coordinates satisfies this equation.

An equation of the form x+√(y(2r-y))=r arccos (r-y)/r specifies in Cartesian coordinates a cycloid - a trajectory that is described by a point on a circle with radius r. In this case, the circle does not follow the abscissa axis, but rolls. What kind of figure this produces, see Figure 1.

A figure whose coordinates of points are given by the following equations:
x=(R+r) cosφ - rcos (R+r)/r φ
y=(R+r) sinφ - rsin (R-r)/r φ,
called an epicycloid. It is the trajectory described by a point on a circle with radius r. This circle rolls along another circle with radius R on the outside. That is an epicycloid, see Figure 2.

Perimeter is one of the mathematical, or more precisely, geometric terms, used mainly to calculate the sides of a figure.

From our article you will learn what perimeter is and how it is measured using the example of basic geometric shapes.

Definition of perimeter

The perimeter is the total length of all sides or the circumference of a figure. The perimeter is denoted by the capital letter “P”, and it can be measured in different units of length, such as millimeters (mm), centimeters (cm), meters (m), etc. For different shapes, there are different formulas for finding the perimeter. Below we will give several examples of how to find out the perimeter of a rectangle and some other shapes.

Measuring the perimeter

If you need to find out the perimeter of a complex figure (such figures include figures with uneven lines), then for this you will need a rope or thread. Using these things, you need to describe the exact outline of the figure, and in order not to get confused, you can make marks on the rope with a pencil. Or you can simply cut it, and then attach all the parts to the ruler. Thus, you will find out what the perimeter of almost any complex figure is.

There is another device for calculating the perimeter of complex figures: it is called a curvimeter (roller rangefinder). With its help, you need to place the roller at any point of the figure and describe the contour of the figure with the roller. The resulting number will be equal to the perimeter. You can learn about finding the perimeter of other geometric shapes from our article. Well, we’ll tell you about several more ways to change the perimeter for different shapes.

Circle, square, equilateral triangle

Let's also look at how to find out the perimeter of a circle. This is quite simple: you just need to determine the circumference, and this can be done by multiplying the radius “r” by the number π≈3.14 and then by 2 (P=L=2∙π∙r).

In this lesson we will introduce a new concept - the perimeter of a rectangle. We will formulate a definition of this concept and derive a formula for its calculation. We will also repeat the combinational law of addition and the distributive law of multiplication.

In this lesson we will learn about the perimeter of a rectangle and its calculation.

Consider the following geometric figure (Fig. 1):

Rice. 1. Rectangle

This figure is a rectangle. Let's remember what distinctive features of a rectangle we know.

A rectangle is a quadrilateral with four right angles and equal sides.

What in our life can have a rectangular shape? For example, a book, a table top or a plot of land.

Consider the following problem:

Task 1 (Fig. 2)

The builders needed to put up a fence around the plot of land. The width of this section is 5 meters, the length is 10 meters. What length of fence will the builders get?

Rice. 2. Illustration for problem 1

The fence is placed along the boundaries of the site, therefore, to find out the length of the fence, you need to know the length of each side. This rectangle has equal sides: 5 meters, 10 meters, 5 meters, 10 meters. Let's create an expression for calculating the length of the fence: 5+10+5+10. Let's use the commutative law of addition: 5+10+5+10=5+5+10+10. This expression contains sums of identical terms (5+5 and 10+10). Let's replace the sums of identical terms with products: 5+5+10+10=5·2+10·2. Now let's use the distributive law of multiplication relative to addition: 5·2+10·2=(5+10)·2.

Let's find the value of the expression (5+10)·2. First we perform the action in brackets: 5+10=15. And then we repeat the number 15 twice: 15·2=30.

Answer: 30 meters.

Perimeter of a rectangle- the sum of the lengths of all its sides. Formula for calculating the perimeter of a rectangle: , here a is the length of the rectangle, and b is the width of the rectangle. The sum of length and width is called semi-perimeter. To obtain the perimeter from the semi-perimeter, you need to increase it by 2 times, that is, multiply by 2.

Let's use the formula for the perimeter of a rectangle and find the perimeter of a rectangle with sides 7 cm and 3 cm: (7 + 3) 2 = 20 (cm).

The perimeter of any figure is measured in linear units.

In this lesson we learned about the perimeter of a rectangle and the formula for calculating it.

The product of a number and the sum of numbers is equal to the sum of the products of the given number and each of the terms.

If the perimeter is the sum of the lengths of all sides of the figure, then the semi-perimeter is the sum of one length and one width. We find the semi-perimeter when we work according to the formula for finding the perimeter of a rectangle (when we perform the first action in parentheses - (a+b)).

Bibliography

1. Alexandrova E.I. Mathematics. 2nd grade. - M.: Bustard, 2004.
2. Bashmakov M.I., Nefedova M.G. Mathematics. 2nd grade. - M.: Astrel, 2006.
3. Dorofeev G.V., Mirakova T.I. Mathematics. 2nd grade. - M.: Education, 2012.

1. Festival.1september.ru ().
2. Nsportal.ru ().
3. Math-prosto.ru ().

Homework

1. Find the perimeter of a rectangle whose length is 13 meters and width is 7 meters.
2. Find the semi-perimeter of a rectangle if its length is 8 cm and width is 4 cm.
3. Find the perimeter of a rectangle if its semi-perimeter is 21 dm.