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. In it, the value of the cosine of the 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.
Perimeter figure is the length of all its sides. Not all shapes have a perimeter, for example, a ball has no perimeter. Standard designation perimeter in mathematics - letter P
Perimeter of a square
Let the length of the side of the square be a. A square has four equal sides, so perimeter of the square is P = a + a + a + a or:
Perimeter of a rectangle
Let the lengths of the sides of the rectangle be a and b.
The length of all its sides is P = a + b + a + b or:
Parallelogram perimeter
Let the lengths of the sides of the parallelogram be a and b
The length of all its sides is P = a + b + a + b, so the perimeter of the parallelogram is:
As you can see, the perimeter of the parallelogram is equal to the perimeter of the rectangle.
Perimeter of an isosceles trapezoid
Let the lengths of the parallel sides of the trapezoid a and b, and the lengths of the other two sides be equal to c (As you know, an isosceles trapezoid has two equal sides).
P = a + b + c + c = a + b + 2c
Perimeter of an equilateral triangle
As you know, an equilateral triangle has 3 equal sides. If the side length is a, then the formula for finding the perimeter is P = a + a + a
Perimeter of the box
A parallelepiped is a prism, all sides of which are parallelograms. ( cuboid is a figure whose sides are rectangles.)
If the sides of the base have lengths a and b then the perimeter of the base is P = 2a + 2b . Each box has two bases, so the perimeter of the two bases is (2a + 2b).2 = 4a + 4b . As we know, the parameter is the sum of all sides. So we have to add four times c
P = 4a + 4b + 4c
cube perimeter
A cube is a parallelepiped, all sides of which are squares (all sides are equal).
Then, the perimeter of a cube is the number of sides * length.
Each cube has 12 sides.
Then, the formula for finding the perimeter of a cube is:
Where a is the length of its side.
How to find the perimeter of various geometric shapes
Having trouble understanding how to find the perimeter of various geometric shapes? The business site comes to your rescue by making geometry easier than ever! Pleasure Fact The perimeter or circumference of the earth is 24,901 miles, i. e. almost 40.075 km! In mathematics, geometry, shapes, sizes, relative position, three-dimensional orientation of figures in space are considered. It deals with the three basic dimensions of figures: area, volume, and perimeter.
Area is a measure of the extent of a two-dimensional figure or shape; a surface can be described as the extent of an object's surface. It is a measure in 3D space near an object.
The perimeter can simply be described as the length of a path that surrounds a two-dimensional shape. In other words, it is the distance around the shape. Let's now take a look at How to find the perimeter of various geometric shapes.
Index
Square
Rectangle
Circle
Semicircle
Sector
Triangle
Trapezoidal
Polygon
Square
A square is a quadrilateral that has all four sides and four angles equal (all 90°).
Example: To find the perimeter of a square with a side of 5 cm, we use the formula shown in Fig.
P = A + A + A + A
P = 5 + 5 + 5 + 5
P = 20 cm
The same formula can be used to calculate the perimeter of a rhombus.
Back to index
Rectangle 
A rectangle is a quadrilateral that has all four angles equal (all 90°). Opposite sides of a rectangle are equal (whereas on adjacent sides No).
Example: To find the perimeter of a rectangle, we use the formula shown in Fig.
l = 15 cm
b = 25 cm
P = 2 (15 + 25)
P = 2 (40)
R = 80 cm
You can use the same formula to find the perimeter of a parallelogram.
Back to index
Circle 
A circle can be described as a set of points equidistant from a particular point (known as the center). The perimeter of a circle is called a circle, denoted c.
Example: find the circumference of a circle, we use the formula shown in Fig..
If C = 2πR and πd
C = 2 x 3.14 x 7 or 3.14 x 14
C = 43.96 cm
Back to index
SEMICIRCLE 
A semicircle, in other words, half a circle, its perimeter will be half of this circle.
Example: To find the perimeter of a semicircle, we use the formula shown in Fig.
p = 7 cm or D = 14 cm (d = p + p)
P \u003d πR and πd / 2
R = 2 x 3.14 x 7 or 3.14 x 14/2
P = 21.98 cm
Back to index
Sector 
A sector can be described as part of a circle.
Example: To find the perimeter of a sector, we use the formula shown in Fig.
ϴ = 60°
p = 7 cm
P \u003d 60/360 X 2 X 3. 14 x 7
R = 7.33 cm
Back to index
Triangle
A triangle is a polygon that has three sides and three vertices. Let's consider three cases in order to determine its perimeter.
one. When all three sides are known.
To find the perimeter of a triangle, we use the formula shown in Fig.
a = 14 cm
b = 16 cm
c = 15 cm
P = 14 + 16 + 15
P = 45 cm 
b. For a right triangle if its hypotenuse is unknown.
To find the perimeter right triangle, we use the formula shown in Fig.
B = 3 cm
h = 4 cm
P \u003d b + h + √ B2 + h 2
P \u003d 3 + 4 + √ 32 + 4 2
P = 3 + 4 + 5
P = 12 cm
If any other side is unknown, one can use the Pythagorean formula to find the side first and then calculate the perimeter. 
With. For any other triangle, when only two sides and an angle are known.
First of all we need to find the length of the side using the law of cosines,
When A, B, and C are the lengths of the sides of a triangle, and a, b, and C have opposite angles of sides A, B, and C, respectively, we can find the length of the unknown side (say, c) by the formula:
C2 \u003d a 2 + B 2 - in 2. b because (c)
For example
A = 4 cm
B=2 cm
C2 \u003d 4 2 + 2 2 - 2 4. 2 cos (45)
C2 = 16 + 4 - 2 (0.876)
C2 = 20 - 1.752
C2 = 18.284
c = 4. 272 cm
P = A + B + C
P = 4 + 2 + 4.272
P = 10.272 cm
Back to index
TRAPEZOID
A trapezoid is a quadrilateral with at least one pair of parallel lines. The parallel lines are called the bases of the trapezoid, and the other side is not known as the legs of the trapezoid. The distance between parallel lines is called the height of the trapezoid.
Let's look at three different scenarios to find the perimeter.
one. When all parties know.
A = 4 cm
b = 16 cm
c = 5 cm
d = 8 cm
P = 4 + 16 + 5 + 8
P = 33 cm 
b. When its sides (legs) are unknown.
To find the perimeter of a trapezoid, we use the formula shown in Fig.
b = 16 cm
h = 3 cm
d = 8 cm
P = b + d + h
1
+
1
Sin(S)
Sin(A)
P = 16 + 8 + 3
1
+
1
Sin(53)
Sin(45)
P = 16 + 8 + 33.3
P = 57.3 cm 
With. When one of the base and height are unknown.
Imagine if we were to cut the trapezoid from two sides in such a way that the lengths of the bases are equal, and when we join the cut part, we get a triangle, as shown in the figure.
When ∠ and ∠c are equal; all three angles are 60°. This triangle is an equilateral triangle, and hence when the length of a side is added to the base, we get the length of the larger base.
When the angles are equal; the sum of the angles subtracted by 180°.
The area of this triangle can be calculated using the formula
A \u003d ½ X X X sin (B)
Find the perimeter of a trapezoid,
A = 4 cm
c = 6 cm
d = 11 cm
∠ a = 53°
∠ c = 65°
∠ B = 78°
Area = ½ x 4 x 6 x sin 78
Area = 6.12 cm2
Triangle base=
Square
½ x x sin(s)
Base =
6. 12
½ x 4 x sin(65)
Base =
6. 12
2 x 0.826
Base = 3.70 cm
Base of the trapezoid = 11 + 3.70 = 14.70 cm
Now we have the sides and base of the trapezoid, we can find the perimeter.
P = 14. 7 + 4 + 6 + 11
P = 35.7 cm
Back to index
Polygon
Any closed figure, where the segments do not intersect with each other, leads to a polygon. The sum of the interior angles of a polygon is always 360°, and they are named according to the number of sides they have.
one. A regular polygon has all equal sides, so when the number of sides and the length of each side is known, the perimeter of the polygon can be calculated using the formula shown in Fig.
Example: If a hexagon has sides 5 cm long, its perimeter can be calculated as shown below.
n = 6 (a hexagon has six sides)
c = 5 cm
P = 6 x 5
R = 30 cm 
b. When the length of the side of the polygon is not known, then its perimeter can be calculated using the formula below.
X = 2 x x Tan (180/p)
Here is a-apothem.
Apothem is a segment from the center of the polygon to the middle of the side.
S = 2 x R x Tan (180/p)
R-radius.
Distance from center regular polygon to any peak.
Example: on a 4 cm apothem hexagon, its side can be calculated as shown below.
c = 2 x 4 x Tan (180/6)
x = 8 x Tan (30)
s = 8 x 0.58
s = 4.62 cm
P = 6 x 4.62 = 27.71 cm
For a hexagon with a radius of 4 cm, its side can be calculated as shown below.
x = 2 x 4 x sin (180/6)
s = 8 x sin (30)
s = 8 x 0.5
s = 4.00 cm
P = 6 x 4. 00 = 24 cm 
With. For an irregular polygon, if all its sides are equal, we can calculate its perimeter by simply adding the lengths of all its sides.
Example: an irregular polygon with six sides
C1 = 8 cm
C2 = 6 cm
C3 = 4 cm
C4=7cm
C5 = 5 cm
C6 = 4 cm
P \u003d C1 + C2 + C3 + C4 + C5 + C6
P \u003d 8 + 6 + 4 + 7 + 5 + 4
P = 36 cm
Back to index
We know geometry can be a little tricky at first (trust us, we know), but keep practicing and you will surely get better with every try.
The ability to find the perimeter of a rectangle is very important for solving many geometric problems. Below is how to find the perimeter of different rectangles.
How to find the perimeter of a regular rectangle
A regular rectangle is a quadrilateral whose parallel sides are equal and all angles = 90º. There are 2 ways to find its perimeter:
Add up all sides.
Calculate the perimeter of a rectangle, if its width is 3 cm and its length is 6.
Solution (sequence of actions and reasoning):
- Since we know the width and length of the rectangle, finding its perimeter is not difficult. The width is parallel to the width, and the length is the length. Thus, in a regular rectangle, there are 2 widths and 2 lengths.
- Add up all sides (3 + 3 + 6 + 6) = 18 cm.
Answer: P = 18 cm.
The second way is as follows:
You need to add the width and length, and multiply by 2. The formula for this method is as follows: 2 × (a + b), where a is the width, b is the length.
As part of this task, we get the following solution:
2x(3 + 6) = 2x9 = 18.
Answer: P = 18.
How to find the perimeter of a rectangle - square
A square is a regular quadrilateral. Correct because all its sides and angles are equal. There are two ways to find its perimeter:
- Add up all of its sides.
- Multiply its side by 4.
Example: Find the perimeter of a square if its side = 5 cm.
Students learn how to find the perimeter in elementary 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. In it, the value of the cosine of the right angle becomes equal to zero, which means that the last term simply disappears.
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.
Determining the perimeter and area of geometric shapes is an important task that arises when solving many practical or everyday problems. If you need to paste wallpaper, install a fence, calculate the consumption of paint or tiles, then you will definitely have to deal with geometric calculations.
To solve the listed everyday issues, you will need to work with a variety of geometric shapes. We present you a catalog of online calculators that allow you to calculate the parameters of the most popular plane figures. Let's consider them.
Circle
Special cases
A quadrilateral with equal sides. A parallelogram becomes a rhombus if its diagonals intersect at 90 degrees and are bisectors of their angles.
It is a parallelogram with right angles. In addition, a parallelogram is considered a rectangle if its sides and diagonals meet the conditions of the Pythagorean theorem.
It is a parallelogram in which all sides are equal and all angles are equal. The diagonals of a square completely repeat the properties of the diagonals of a rectangle and a rhombus, which makes the square a unique figure that is characterized by maximum symmetry.
Polygon
A regular polygon is a convex figure on a plane that has equal sides and equal angles. Polygons have their own names depending on the number of sides:
- - pentagon;
- - hexagon;
- eight - octagon;
- twelve - dodecagon.
And so on. Geometers joke that a circle is a polygon with an infinite number of angles. Our calculator is programmed to determine the perimeters and areas of regular polygons only. It uses general formulas for all regular polygons. To calculate the perimeter, the formula is used:
where n is the number of sides of the polygon, a is the length of the side.
To determine the area, the expression is used:
S = n/4 × a^2 × ctg(pi/n).
Substituting the appropriate n, we can find a formula for any regular polygon, which also includes an equilateral triangle and a square.
Polygons are very common in real life. So the shape of a pentagon is the building of the US Department of Defense - the Pentagon, a hexagon - honeycombs or snowflake crystals, an octagon - road signs. In addition, many protozoa, such as radiolarians, have the shape of regular polygons.
Real life examples
Let's look at a couple of examples of using our calculator in real-life calculations.
Fence painting
Surface painting and paint calculation are some of the most obvious everyday tasks that require minimal mathematical calculations. If we need to paint a fence that is 1.5 meters high and 20 meters long, how many cans of paint do we need? To do this, you need to find out the total area of \u200b\u200bthe fence and the consumption of paints and varnishes per 1 square meter. We know that enamel consumption is 130 grams per meter. Now let's determine the area of the fence using the calculator to calculate the area of the rectangle. It will be S = 30 square meters. Naturally, we will paint the fence on both sides, so the area for painting will increase to 60 squares. Then we need 60 × 0.13 = 7.8 kilograms of paint, or three standard cans of 2.8 kilograms.
Fringe trim
Tailoring is another industry that requires extensive geometric knowledge. Suppose we need to fringe a scarf, which is an isosceles trapezoid with sides of 150, 100, 75 and 75 cm. To calculate the fringe consumption, we need to know the perimeter of the trapezoid. This is where the online calculator comes in handy. Enter this cell data and get the answer:
Thus, we need 4 m of fringe to finish the scarf.
Conclusion
Flat figures make up the real world around. We often asked ourselves at school the question, will geometry be useful to us in the future? The above examples show that mathematics is constantly used in everyday life. And if the area of a rectangle is familiar to us, then calculating the area of \u200b\u200bthe dodecagon can be a difficult task. Use our catalog of calculators to solve school assignments or everyday problems.
, 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: the inner and outer area relative to the figure to which it belongs:
Perimeter
The perimeter is a closed boundary of a flat geometric figure that separates its inner area from the 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 a polygon is always equal to the sum of the lengths 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 region, 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 of 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 |
One of the basic concepts of mathematics is the perimeter of a rectangle. There are many problems on this topic, the solution of which cannot do without the perimeter formula and the skills to calculate it.
Basic concepts
A rectangle is a quadrilateral in which all angles are right and opposite sides are pairwise equal and parallel. In our life, many figures are in the shape of a rectangle, for example, the surface of a table, a notebook, and so on.
Consider an example: a fence must be placed along the boundaries of the land. In order to find out the length of each side, you need to measure them.
Rice. 1. Land plot in the shape of a rectangle.
The land plot has sides with a length of 2 m, 4 m, 2 m, 4 m. Therefore, in order to find out the total length of the fence, you must add the lengths of all sides:
2+2+4+4= 2 2+4 2 =(2+4) 2 =12 m.
It is this value that is generally called the perimeter. Thus, to find the perimeter, you need to add all the sides of the figure. The letter P is used to designate the perimeter.
To calculate the perimeter of a rectangular figure, you do not need to divide it into rectangles, you need to measure only all sides of this figure with a ruler (tape measure) and find their sum.
The perimeter of a rectangle is measured in mm, cm, m, km, and so on. If necessary, the data in the task are converted into the same measurement system.
The perimeter of a rectangle is measured in various units: mm, cm, m, km, and so on. If necessary, the data in the task is converted into one system of measurement.
Shape Perimeter Formula
If we take into account the fact that opposite sides of a rectangle are equal, then we can derive the formula for the perimeter of a rectangle:
$P = (a+b) * 2$, where a, b are the sides of the figure.
Rice. 2. Rectangle, with opposite sides marked.
There is another way to find the perimeter. If the task is given only one side and the area of \u200b\u200bthe figure, you can use to express the other side through the area. Then the formula will look like this:
$P = ((2S + 2a2)\over(a))$, where S is the area of the rectangle.
Rice. 3. Rectangle with sides a, b.
Exercise : Calculate the perimeter of a rectangle if its sides are 4 cm and 6 cm.
Solution:
We use the formula $P = (a+b)*2$
$P = (4+6)*2=20 cm$
Thus, the perimeter of the figure is $P = 20 cm$.
Since the perimeter is the sum of all the sides of a figure, the semi-perimeter is the sum of only one length and width. Multiply the semi-perimeter by 2 to get the perimeter.
Area and perimeter are the two basic concepts for measuring any figure. They should not be confused, although they are related. If you increase or decrease the area, then, accordingly, its perimeter will increase or decrease.
What have we learned?
We have learned how to find the perimeter of a rectangle. And also got acquainted with the formula for its calculation. This topic can be encountered not only when solving math problems but also in real life.
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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 .
