, broken line, etc.:
If you look closely at all these figures, you can select two of them, which are formed by closed lines (a circle and a triangle). These figures have a kind of border separating what is inside from what is outside. That is, the boundary divides the plane into two parts: internal and outer area regarding the figure to which it refers:
Perimeter
The perimeter is the closed boundary of a plane geometric figure separating its inner region from its outer one.
Any closed geometric figure has a perimeter:
In the figure, the perimeters are marked with a red line. Note that the circumference of a circle is often referred to as the length.
The perimeter is measured in units of length: mm, cm, dm, m, km.
For all polygons, finding the perimeter is reduced to adding the lengths of all sides, that is, the perimeter of the polygon is always is equal to the sum the length of its sides. When calculating the perimeter, it is often denoted by a capital Latin letter P:
Square
Area is the part of the plane occupied by a closed flat geometric figure.
Any flat closed geometric figure has a certain area. In the drawings, the area of geometric shapes is the inner area, that is, that part of the plane that is inside the perimeter.
measure area figures - means to find how many times another figure is placed in a given figure, taken as a unit of measurement. Usually, a square is taken as a unit of area measurement, in which the side is equal to the unit of length measurement: millimeter, centimeter, meter, etc.
The figure shows a square centimeter. - a square with each side 1 cm long:
Area is measured in square units ah measuring length. Area units include: mm 2, cm 2, m 2, km 2, etc.
Square units conversion table
| mm 2 | cm 2 | dm 2 | m 2 | ar (weave) | hectare (ha) | km 2 | |
|---|---|---|---|---|---|---|---|
| mm 2 | 1 mm 2 | 0.01 cm2 | 10 -4 dm 2 | 10 -6 m 2 | 10 -8 ar | 10 -10 ha | 10 -12 km 2 |
| cm 2 | 100 mm 2 | 1 cm 2 | 0.01 dm 2 | 10 -4 m 2 | 10 -6 are | 10 -8 ha | 10 -10 km 2 |
| dm 2 | 10 4 mm 2 | 100 cm 2 | 1 dm 2 | 0.01 m2 | 10 -4 ar | 10 -6 ha | 10 -8 km 2 |
| m 2 | 10 6 mm 2 | 10 4 cm 2 | 100 dm 2 | 1 m 2 | 0.01 are | 10 -4 ha | 10 -6 km 2 |
| ar | 10 8 mm 2 | 10 6 cm 2 | 10 4 dm 2 | 100 m2 | 1 are | 0.01 ha | 10 -4 km 2 |
| ha | 10 10 mm 2 | 10 8 cm 2 | 10 6 dm 2 | 10 4 m 2 | 100 are | 1 ha | 0.01 km2 |
| km 2 | 10 12 mm 2 | 10 10 cm 2 | 10 8 dm 2 | 10 6 m 2 | 10 4 ar | 100 ha | 1 km 2 |
| 10 4 = 10 000 | 10 -4 = 0,000 1 |
| 10 6 = 1 000 000 | 10 -6 = 0,000 001 |
| 10 8 = 100 000 000 | 10 -8 = 0,000 000 01 |
| 10 10 = 10 000 000 000 | 10 -10 = 0,000 000 000 1 |
| 10 12 = 1 000 000 000 000 | 10 -12 = 0,000 000 000 001 |
Knowledge of how to find the perimeter, students receive in primary school. Then this information is constantly used throughout the course of mathematics and geometry.
Theory common to all figures
The parties are usually denoted in Latin letters. Moreover, they can be designated as segments. Then you will need two letters for each side and written in large letters. Or enter the designation with one letter, which will necessarily be small.
Letters are always chosen alphabetically. For a triangle, they will be the first three. The hexagon will have 6 of them - from a to f. This is useful for entering formulas.
Now about how to find the perimeter. It is the sum of the lengths of all sides of the figure. The number of terms depends on its type. The perimeter is denoted by the Latin letter P. The units of measurement are the same as those given for the sides.
Perimeter formulas for different shapes
For a triangle: P \u003d a + b + c. If it is isosceles, then the formula is converted: P \u003d 2a + c. How to find the perimeter of a triangle if it is equilateral? This will help: P \u003d 3a.
For an arbitrary quadrilateral: P=a+b+c+d. Its special case is the square, the perimeter formula: P=4a. There is also a rectangle, then the following equality is required: P \u003d 2 (a + b).
What if you don't know the length of one or more sides of a triangle?
Use the cosine theorem if there are two sides among the data and the angle between them, which is denoted by the letter A. Then, before finding the perimeter, you will have to calculate the third side. For this, the following formula is useful: c² \u003d a² + b² - 2 av cos (A).
A special case of this theorem is the one formulated by Pythagoras for a right triangle. It contains the value of the cosine right angle becomes zero, which means that the last term simply vanishes.
There are situations when you can find out how to find the perimeter of a triangle on one side. But at the same time, the angles of the figure are also known. Here the sine theorem comes to the rescue, when the ratios of the lengths of the sides to the sines of the corresponding opposite angles are equal.
In a situation where the perimeter of a figure needs to be found by area, other formulas will come in handy. For example, if the radius of the inscribed circle is known, then in the question of how to find the perimeter of a triangle, the following formula is useful: S \u003d p * r, here p is the semi-perimeter. It must be derived from this formula and multiplied by two.
Task examples
First condition. Find the perimeter of a triangle whose sides are 3, 4 and 5 cm.
Solution. You need to use the equality that is indicated above, and simply substitute the data in the value task into it. The calculations are easy, they lead to the number 12 cm.
Answer. The perimeter of a triangle is 12 cm.
Second condition. One side of the triangle is 10 cm. It is known that the second is 2 cm larger than the first, and the third is 1.5 times larger than the first. It is required to calculate its perimeter.
Solution. In order to find out, you need to count two sides. The second is defined as the sum of 10 and 2, the third is equal to the product of 10 and 1.5. Then it remains only to count the sum of three values: 10, 12 and 15. The result will be 37 cm.
Answer. The perimeter is 37 cm.
Third condition. There is a rectangle and a square. One side of the rectangle is 4 cm, and the other is 3 cm longer. It is necessary to calculate the value of the side of the square if its perimeter is 6 cm less than that of the rectangle.
Solution. The second side of the rectangle is 7. Knowing this, it is easy to calculate its perimeter. The calculation gives 22 cm.
To find out the side of the square, you must first subtract 6 from the perimeter of the rectangle, and then divide the resulting number by 4. As a result, we have the number 4.
Answer. The side of the square is 4 cm.
Surely each of us learned at school such an important component of geometry as the perimeter. Finding the perimeter is simply necessary to solve 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.
- 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 the problem:
Let's say 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.
- A trapezoid is a quadrilateral whose two opposite sides are parallel but not equal. The perimeter of a trapezoid is the sum of all four of its sides:
P = X+Y+Z+W, where X, Y, Z, W are the sides of the figure.
An example of solving the problem:
Let's say 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.
- 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 the problem:
Let's say that the radius r of our circle is 5 cm, then the diameter d will be 2 * 5 cm = 10 cm. It is known that π = 3.14. So, substituting these values into our formula, we get - P = 2 * 5 cm * 3.14 = 31.4 cm.
- If you need to find the perimeter of a triangle, then you may run into a number of problems while doing this, since triangles can have very different shapes. For example, there are acute, obtuse, isosceles, right or 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 of finding 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 the angle or the length of the height of a particular triangle. This will significantly complicate the task, but will not make its solution unrealistic. How to find the perimeter of a triangle, no matter what shape it is, you can read "".
- The perimeter of such a figure 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!
In the next test tasks Find the perimeter of the figure shown in the figure.
There are many ways to find the perimeter of a shape. You can transform the original shape in such a way that the perimeter of the new shape can be easily calculated (for example, change to a rectangle).
Another solution is to look for the perimeter of the figure directly (as the sum of the lengths of all its sides). But in this case, one cannot rely only on the drawing, but find the lengths of the segments based on the data of the problem.
I want to warn you: in one of the tasks, among the proposed answers, I did not find the one that turned out for me.
c) .
Let's move the sides of the small rectangles from the inner area to the outer one. As a result, the large rectangle is closed. Formula for Finding the Perimeter of a Rectangle
In this case, a=9a, b=3a+a=4a. Thus P=2(9a+4a)=26a. To the perimeter of the large rectangle we add the sum of the lengths of four segments, each of which is equal to 3a. As a result, P=26a+4∙3a= 38a .
c) .
After transferring the inner sides of the small rectangles to the outer area, we get a large rectangle, the perimeter of which is P=2(10x+6x)=32x, and four segments, two of x length, two of 2x length.
Total, P=32x+2∙2x+2∙x= 38x .
?) .
Let's move 6 horizontal "steps" from the inside to the outside. The perimeter of the resulting large rectangle is P=2(6y+8y)=28y. It remains to find the sum of the lengths of the segments inside the rectangle 4y+6∙y=10y. Thus, the perimeter of the figure is P=28y+10y= 38y .
D) .
Let's move the vertical segments from the inner area of the figure to the left, to the outer area. To get a big rectangle, move one of the 4x lengths to the bottom left corner.
We find the perimeter of the original figure as the sum of the perimeter of this large rectangle and the lengths of the remaining three segments P=2(10x+8x)+6x+4x+2x= 48x .
e) .
Moving the inner sides of the small rectangles to the outer area, we get a large square. Its perimeter is P=4∙10x=40x. To get the perimeter of the original figure, you need to add the sum of the lengths of eight segments, each 3x long, to the perimeter of the square. Total, P=40x+8∙3x= 64x .
b) .
Let's move all horizontal "steps" and vertical upper segments to the outer area. The perimeter of the resulting rectangle is P=2(7y+4y)=22y. To find the perimeter of the original figure, you need to add to the perimeter of the rectangle the sum of the lengths of four segments, each with a length of y: P=22y+4∙y= 26y .
D) .
Move all horizontal lines from the inner area to the outer area and move the two vertical outer lines in the left and right corners, respectively, z to the left and right. As a result, we get a large rectangle, the perimeter of which is P=2(11z+3z)=28z.
The perimeter of the original figure is equal to the sum of the perimeter of the large rectangle and the lengths of six segments in z: P=28z+6∙z= 34z .
b) .
The solution is completely similar to the solution of the previous example. After transforming the figure, we find the perimeter of the large rectangle:
P=2(5z+3z)=16z. To the perimeter of the rectangle we add the sum of the lengths of the remaining six segments, each of which is equal to z: P=16z+6∙z= 22z .
Geometry, if I'm not mistaken, in my time was studied from the fifth grade and the perimeter was and is one of the key concepts. So, perimeter is the sum of the lengths of all sides (denoted by the Latin letter P). In general, this term is interpreted in different ways, for example,
- the total length of the border of the figure,
- the length of all its sides,
- the sum of the lengths of its faces,
- the length of the bounding line,
- the sum of all the lengths of the sides of a polygon
Different shapes have their own formulas for determining the perimeter. To understand the meaning itself, I propose to independently deduce a few simple formulas:
- for a square
- for a rectangle
- for a parallelogram
- for cube
- for a box
Perimeter of a square
For example, let's take the simplest - the perimeter of a square.
All sides of a square are equal. Let one side be called "a" (as well as the other three), then
P = a + a + a + a
or more compact notation
Perimeter of a rectangle
Let's complicate the task and take a rectangle. In this case, it is no longer possible to say that all sides are equal, so let the lengths of the sides of the rectangle be equal to a and b.
Then the formula will look like this:
P = a + b + a + b
Parallelogram perimeter
A similar situation will be with a parallelogram (see the perimeter of the rectangle)
cube perimeter
What to do if we are dealing with a three-dimensional figure? For example, take a cube. A cube has 12 sides and they are all equal. Accordingly, the perimeter of a cube can be calculated as follows:
Perimeter of the box
Well, to fix the material, we calculate the perimeter of the parallelepiped. Here it is necessary to think a little. Let's do it together. As we know, a cuboid is a figure whose sides are rectangles. Each parallelepiped has two bases. Let's take one of the bases and look at its sides - they have lengths a and b. Accordingly, the perimeter of the base is P = 2a + 2b. Then the perimeter of the two bases is
(2a + 2b) * 2 = 4a + 4b
But we also have a "c" side. So the formula for calculating the perimeter of a parallelepiped will look like this:
P = 4a + 4b + 4c
As you can see from the examples above, all that needs to be done to determine the perimeter of a shape is to find the length of each of the sides, and then add them up.
In conclusion, I would like to note that not every figure has a perimeter. For example, A sphere has no perimeter.
