It is interesting that many years ago such a branch of mathematics as "geometry" was called "surveying". And how to find the perimeter and area has been known for a long time. For example, they say that the very first calculators of these two quantities are the inhabitants of Egypt. Thanks to this knowledge, they were able to build structures known today.
The ability to find area and perimeter can be useful in Everyday life. In everyday life, these values \u200b\u200bare used when it is necessary to paint something, plant or process a garden, glue wallpaper in a room, etc.
Perimeter
Most often, you need to find out the perimeter of polygons or triangles. To determine this value, it is enough just to know the lengths of all sides, and the perimeter is their sum. Finding the perimeter if the area is known is also possible.
Triangle
If you need to know the perimeter of a triangle, to calculate it, you should apply the following formula P \u003d a + b + c, where a, b, c are the sides of the triangle. In this case, all sides of an ordinary triangle on the plane are summed up.
A circle
The perimeter of a circle is usually called the circumference of a circle. To find out this value, you must use the formula: L \u003d π * D \u003d 2 * π * r, where L is the circumference, r is the radius, D is the diameter, and the number π, as you know, is approximately equal to 3.14.
square, rhombus
The formulas for the perimeters of a square and a rhombus are the same, because for one figure and for the other, all sides are equal. Since a square and a rhombus have equal sides, they (the sides) can be denoted by one letter "a". It turns out that the perimeter of a square and a rhombus is equal to:
- P \u003d a + a + a + a or P \u003d 4a
Rectangle, parallelogram
A rectangle and a parallelogram have the same opposite sides, so they can be denoted by two different letters "a" and "b". The formula looks like this:
- P \u003d a + b + a + b \u003d 2a + 2b. The deuce can be taken out of brackets, and the following formula will turn out: P \u003d 2 (a + b)
Trapeze
A trapezoid has different sides, so they are denoted by different letters of the Latin alphabet. In this regard, the formula for the perimeter of a trapezoid looks like this:
- P = a + b + c + d Here all sides are added together.
Square
Area - that part of the figure, which is enclosed within its contour.
Rectangle
To calculate the area of a rectangle, you need to multiply the value of one side (length) by the value of the other (width). If the length and width values are denoted by the letters "a" and "b", then the area is calculated by the formula:
- S = a*b
Square
As you already know, the sides of a square are equal, so to calculate the area, you can simply take one side into a square:
- S \u003d a * a \u003d a 2
Rhombus
The formula for finding the area of a rhombus has a slightly different form: S \u003d a * h a, where h a is the length of the height of the rhombus, which is drawn to the side.
In addition, the area of a rhombus can be found by the formulas:
- S \u003d a 2 * sin α, while a is the side of the figure, and the angle α is the angle between the sides;
- S \u003d 4r 2 / sin α, where r is the radius of the circle inscribed in the rhombus, and the angle α is the angle between the sides.
A circle
The area of a circle is also easily recognized. To do this, you can use the formula:
- S \u003d πR 2, where R is the radius.
Trapeze
To calculate the area of a trapezoid, you can use this formula:
- S \u003d 1/2 * a * b * h, where a, b are the bases of the trapezoid, h is the height.
Triangle
To find the area of a triangle, use one of several formulas:
- S \u003d 1/2 * a * b sin α (where a, b are the sides of the triangle, and α is the angle between them);
- S \u003d 1/2 a * h (where a is the base of the triangle, h is the height lowered to it);
- S \u003d abc / 4R (where a, b, c are the sides of the triangle, and R is the radius of the circumscribed circle);
- S \u003d p * r (where p is the semi-perimeter, r is the radius of the inscribed circle);
- S= √ (p*(p-a)*(p-b)*(p-c)) (where p is the semi-perimeter, a, b, c are the sides of the triangle).
Parallelogram
To calculate the area of this figure, you must substitute the values in one of the formulas:
- S \u003d a * b * sin α (where a, b are the bases of the parallelogram, α is the angle between the sides);
- S \u003d a * h a (where a is the side of the parallelogram, h a is the height of the parallelogram, which is lowered to side a);
- S = 1/2 *d*D* sin α (where d and D are the diagonals of the parallelogram, α is the angle between them).
Rectangle - P = 2*a + 2*b = 2*3 + 2*6 = 6 + 12 = 18. In this problem, the perimeter coincided in value with the area of the figure.
Square Problem: find the perimeter of a square if its area is 9. Solution: using the square area formula S = a ^ 2, from here find the length of the side a = 3. The perimeter is equal to the sum of the lengths of all sides, therefore, P = 4 * a = 4 * 3 = 12.
Triangle Task: given an arbitrary ABC, the area of \u200b\u200bwhich is equal to 14. Find the perimeter of the triangle if the line drawn from the vertex B divides the base of the triangle into segments of length 3 and 4 cm. S = ½*AC*BE. The perimeter is equal to the sum of the lengths of all sides. Find the length of side AC by adding the lengths AE and EC, AC = 3 + 4 = 7. Find the height of the triangle BE = S*2/AC = 14*2/7 = 4. Consider a right triangle ABE. Knowing AE and BE, you can find the hypotenuse using the Pythagorean formula AB^2 = AE^2 + BE^2, AB = √(3^2 + 4^2) = √25 = 5. Consider right triangle BEC. According to the Pythagorean formula BC^2 = BE^2 + EC^2, BC = √(4^2 + 4^2) = 4*√2. Now the lengths of all sides of the triangle. Find the perimeter from their sum P = AB + BC + AC = 5 + 4*√2 + 7 = 12 + 4*√2 = 4*(3+√2).
CircleProblem: it is known that the area of a circle is 16*π, find its perimeter. Solution: write down the formula for the area of a circle S = π*r^2. Find the radius of the circle r = √(S/π) = √16 = 4. According to the formula, the perimeter is P = 2*π*r = 2*π*4 = 8*π. If we accept that π = 3.14, then P = 8*3.14 = 25.12.
Sources:
- area equals perimeter
All of us once in school begin to study the perimeter of a rectangle. So let's remember how to calculate it and what is the perimeter in general?
The word "perimeter" comes from two Greek words: "peri", which means "around", "about" and "metron", which means "to measure", "to measure". Those. perimeter, translated from Greek means "measurement around."
Instruction
The second definition will sound like this: the perimeter of a rectangle is twice the sum of its length and width.
Related videos
The area of a rectangle is the product of its length times its width. Pemeter is the sum of all sides.
Sources:
A circle is a geometric figure formed from a set of points that are far from the center. circles for an equal distance. Based on the known circles data, there are 2 formulas arising from each other for determining its area.
You will need
- The value of the constant π (equal to 3.14);
- The size of the diameter/radius of a circle.
Instruction
Related videos
A square is a beautiful and simple flat geometric figure. It is a rectangle with equal sides. How to find perimeter square if the length of its side is known?
Instruction
First of all, remember that perimeter is nothing more than the sum of a geometric figure. Considered by us four sides. Moreover, by , all these sides are equal between .
From these premises, it is easy to find perimeter a square – perimeter square side length square multiplied by four:
P \u003d 4a, where a is the length of the side square.
Related videos
Tip 6: How to find the area of a triangle and a rectangle
Triangle and rectangle are two of the simplest flat geometric figures in Euclidean geometry. Within the perimeters formed by the sides of these polygons, there is a certain section of the plane, the area of which can be determined in many ways. The choice of method in each particular case will depend on the known parameters of the figures.
Instruction
Use one of the trigonometric formulas to find the area of a triangle if you know the values of one or more angles in . For example, with a known value of the angle (α) and the lengths of the sides that make it up (B and C), the area (S) can be obtained by the formula S \u003d B * C * sin (α) / 2. And with the values of all angles (α, β and γ) and the length of one side in addition (A), you can use the formula S \u003d A² * sin (β) * sin (γ) / (2 * sin (α)). If, in addition to all angles, (R) of the circumscribed circle is known, then use the formula S=2*R²*sin(α)*sin(β)*sin(γ).
If the angles are not known, then to find the area of a triangle, you can use without trigonometric functions. For example, if (H) drawn from a side that also knows (A), then use the formula S \u003d A * H / 2. And if the lengths of each of the sides (A, B and C) are given, then first find the semi-perimeter p \u003d (A + B + C) / 2, and then calculate the area of \u200b\u200bthe triangle using the formula S \u003d √ (p * (p-A) * (p-B) * (p-C)). If, in addition to (A, B and C), the radius (R) of the circumscribed circle is known, then use the formula S \u003d A * B * C / (4 * R).
To find the area of a rectangle, trigonometric functions can also be used - for example, if the length of its diagonal (C) and the angle that it has on one of the sides (α) are known. In this case, use the formula S=С²*sin(α)*cos(α). And if the lengths of the diagonals (C) and the angle they make up (α) are known, then use the formula S \u003d C² * sin (α) / 2.
When solving, it is necessary to take into account that solving the problem of finding the area of a rectangle only from the length of its sides it is forbidden.
This is easy to verify. Let the perimeter of the rectangle be 20 cm. This will be true if its sides are 1 and 9, 2 and 8, 3 and 7 cm. All these three rectangles will have the same perimeter, equal to twenty centimeters. (1 + 9) * 2 = 20 just like (2 + 8) * 2 = 20 cm.
As you can see, we can choose an infinite number of options the dimensions of the sides of the rectangle, the perimeter of which will be equal to the given value.
The area of rectangles with a given perimeter of 20 cm, but with different sides will be different. For the given example - 9, 16 and 21 square centimeters, respectively.
S 1 \u003d 1 * 9 \u003d 9 cm 2
S 2 \u003d 2 * 8 \u003d 16 cm 2
S 3 \u003d 3 * 7 \u003d 21 cm 2
As you can see, there are an infinite number of options for the area of \u200b\u200ba figure with a given perimeter.
Thus, in order to calculate the area of a rectangle from its perimeter, it is necessary to know either the ratio of its sides or the length of one of them. The only figure that has an unambiguous dependence of its area on the perimeter is a circle. Only for circle and possibly a solution.
In this lesson:
- Task 4. Change the length of the sides while maintaining the area of the rectangle
Task 1. Find the sides of a rectangle from the area
The perimeter of a rectangle is 32 centimeters, and the sum of the areas of the squares built on each of its sides is 260 square centimeters. Find the sides of the rectangle.Decision.
2(x+y)=32
According to the condition of the problem, the sum of the areas of the squares built on each of its sides (squares, respectively, four) will be equal to
2x2+2y2=260
x+y=16
x=16-y
2(16-y) 2 +2y 2 =260
2(256-32y+y2)+2y2=260
512-64y+4y 2 -260=0
4y2 -64y+252=0
D=4096-16x252=64
x1=9
x2=7
Now let's take into account that based on the fact that x+y=16 (see above) at x=9, then y=7 and vice versa, if x=7, then y=9
Answer: The sides of a rectangle are 7 and 9 centimeters
Task 2. Find the sides of a rectangle from the perimeter
The perimeter of a rectangle is 26 cm, and the sum of the areas of the squares built on its two adjacent sides is 89 square meters. see Find the sides of the rectangle.
Decision.
Let's denote the sides of the rectangle as x and y.
Then the perimeter of the rectangle is:
2(x+y)=26
The sum of the areas of the squares built on each of its sides (there are two squares, respectively, and these are the squares of the width and height, since the sides are adjacent) will be equal to
x2+y2=89
We solve the resulting system of equations. From the first equation we deduce that
x+y=13
y=13-y
Now we perform a substitution in the second equation, replacing x with its equivalent.
(13th) 2 +y 2 =89
169-26y+y 2 +y 2 -89=0
2y2 -26y+80=0
We solve the resulting quadratic equation.
D=676-640=36
x1=5
x2=8
Now let's take into account that based on the fact that x+y=13 (see above) at x=5, then y=8 and vice versa, if x=8, then y=5
Answer: 5 and 8 cm
Task 3. Find the area of a rectangle from the proportion of its sides
Find the area of a rectangle if its perimeter is 26 cm and the sides are proportional as 2 to 3.
Decision.
Let us denote the sides of the rectangle by the coefficient of proportionality x.
From where the length of one side will be equal to 2x, the other - 3x.
Then:
2(2x+3x)=26
2x+3x=13
5x=13
x=13/5
Now, based on the data obtained, we determine the area of the rectangle:
2x*3x=2*13/5*3*13/5=40.56 cm2
Task 4. Changing the length of the sides while maintaining the area of a rectangle
Rectangle length increased by 25%. By what percentage should the width be reduced so that its area does not change?Decision.
The area of the rectangle is
S=ab
In our case, one of the factors increased by 25%, which means a 2 = 1.25a. So the new area of the rectangle should be
S 2 \u003d 1.25ab
Thus, in order to return the area of the rectangle to its initial value, then
S2 = S / 1.25
S 2 \u003d 1.25ab / 1.25
Since the new size a cannot be changed, then
S 2 \u003d (1.25a) b / 1.25
1 / 1,25 = 0,8
Thus, the value of the second side must be reduced by (1 - 0.8) * 100% = 20%
Answer: Width should be reduced by 20%.
To find the perimeter and area of a rectangle, you need know the formulas and most importantly - be able to apply them to solve problems - because they are of varying complexity.
Very often, when solving problems of an easy level, it is enough to know the basic formulas and solve them simply by substituting the necessary values.
If the tasks are more complicated and their conditions do not contain the data necessary for the formula, they need to be found using other algebraic operations.
In this case, you can use the following example
you need to find the area of a rectangle if its perimeter is 120 cm, and the ratio of the sides is 2 to 3
at first write an equation to find the sides using the perimeter formula ( P=2(a+b):
2*(2x+3X)=120 solve it, x=12 means the sides are 24 cm and 36 cm and now we substitute the values into the area formula S=ab and find it S=24*36=864 sq.cm.
The area of a rectangle is equal to the product of length and width and is calculated by the formula a * b, where a and b are the sides of the rectangle. The perimeter of a rectangle is equal to the sum of all its sides and is calculated by the formula a+b+a+b.
Finding the area of a rectangle - multiply the length of the rectangle by its width.
Finding the perimeter of a rectangle (the sum of the lengths of all sides) - by simply adding the lengths of all sides, or to the length of the longitudinal side of the rectangle, add the length of the transverse side and multiply the resulting amount by two.
If you imagine that your garden is rectangular and you need to fence the plot, then you will probably have a question, how long will the fence be in order to correctly calculate the consumption of building materials. You add up the lengths of the sides of the fence to find the PERIMETER. If you ask yourself how much land you need to dig in this area, you will have to look for AREA, and for this you will need to multiply the length by the width of the area, because as you know, the opposite sides of a rectangle are equal in pairs. Do not forget that a square is also a rectangle, to find the perimeter of a square, you need to multiply the length by 4, and the area - the length of the side, multiply by itself.
Think back to high school math. So the perimeter of a rectangle is found by the formula of the sum of its two sides multiplied by 2. That is, P \u003d 2 * (a + b), where a and b are the sides of the rectangle. The area, respectively, is found using the formula S=a*b, where a and b are also its sides.
If you do not go into deep details, then finding the area and perimeter of a rectangle is very simple. We denote the sides of such a rectangle in Latin letters: a, b, c and d. Let a = c be the length of the rectangle and b and d be the width of the rectangle.
Rectangle area:
Rectangle Perimeter:
S = a + b + c + d
The perimeter of a rectangle is the length of all its sides. Based on the fact that this figure has four sides, or two pairs, while the opposite sides are equal to each other, we can conclude that it is appropriate to add the values \u200b\u200bof two sides of different sizes and multiply the resulting value by two.
The area is also simple: we simply multiply sides of different sizes.
The area is calculated by multiplying the long side of the rectangle with the short side. And the perimeter is (long side + short side) * 2
You can go on your own simple way finding the area of a rectangle. Namely, multiply the length of the rectangle (usually a) by the width of the rectangle (usually B). But we are looking for the perimeter by adding all sides, or, more simply: 2a + 2b
Rectangle it is a geometric figure, namely a quadrilateral, in which all angles are right. It turns out that the opposite sides are equal to each other.
Perimeter of a rectangle is the sum of the lengths of all sides of the rectangle, or the sum of the length and width multiplied by 2.
Perimeter is the length of all sides of the rectangle, then it is measured in units of length: cm, mm, m, dm, km.
P=AB+CD+AD+BC or P=2*(AB+AD).
Square measured in square units of length: m2, cm2, dm2 and is denoted by the Latin letter S.
To find the area of a rectangle, multiply the length of the rectangle by its width.
The area of a rectangle is calculated by multiplying its length by the width of the resulting product and will be the area.
The perimeter of the rectangle is found by summing the length and width, the resulting sum must also be multiplied by two, this will be the desired perimeter.
If a rectangle has two opposite sides, then we simply multiply them and get the area, add and double and get the perimeter. However, more often in textbooks they ask the most inconsistency - side and perimeter, side and area, side and diagonal. How to proceed in these cases.
This is the ideal task.
Side and diagonal can be specified. In this case, we find the second side according to the Pythagorean theorem - as the second leg in a triangle where the hypotenuse is the diagonal of the rectangle.
As a result, we have the following formulas for finding the perimeter of a rectangle:
And if you simply transform these same formulas, then you get formulas for finding the area in all variants of tasks:
Lesson and presentation on the topic: "Perimeter and area of a rectangle"
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What is a rectangle and a square
Rectangle is a quadrilateral with all right angles. So the opposite sides are equal to each other.
Square is a rectangle with equal sides and angles. It is called a regular quadrilateral.
Quadrilaterals, including rectangles and squares, are denoted by 4 letters - vertices. Latin letters are used to designate vertices: A, B, C, D...
Example. 
It reads like this: quadrilateral ABCD; square EFGH.
What is the perimeter of a rectangle? Formula for calculating the perimeter
Perimeter of a rectangle is the sum of the lengths of all sides of the rectangle, or the sum of the length and width multiplied by 2.The perimeter is indicated by the Latin letter P. Since the perimeter is the length of all sides of the rectangle, the perimeter is written in units of length: mm, cm, m, dm, km.
For example, the perimeter of a rectangle ABCD is denoted as P ABCD, where A, B, C, D are the vertices of the rectangle.
Let's write the formula for the perimeter of quadrilateral ABCD:
P ABCD = AB + BC + CD + AD = 2 * AB + 2 * BC = 2 * (AB + BC)
Example.
A rectangle ABCD is given with sides: AB=CD=5 cm and AD=BC=3 cm.
Let's define P ABCD .
Decision:
1. Let's draw a rectangle ABCD with initial data. 
2. Let's write a formula for calculating the perimeter of this rectangle:
P ABCD = 2 * (AB + BC)
P ABCD=2*(5cm+3cm)=2*8cm=16cm
Answer: P ABCD = 16 cm.
The formula for calculating the perimeter of a square
We have a formula for finding the perimeter of a rectangle.P ABCD=2*(AB+BC)
Let's use it to find the perimeter of a square. Considering that all sides of the square are equal, we get:
P ABCD=4*AB
Example.
Given a square ABCD with a side equal to 6 cm. Determine the perimeter of the square.
Decision.
1. Draw a square ABCD with the original data.
2. Recall the formula for calculating the perimeter of a square:
P ABCD=4*AB
3. Substitute our data into the formula:
P ABCD=4*6cm=24cm
Answer: P ABCD = 24 cm.
Problems for finding the perimeter of a rectangle
1. Measure the width and length of the rectangles. Determine their perimeter. 
2. Draw a rectangle ABCD with sides 4 cm and 6 cm. Determine the perimeter of the rectangle.
3. Draw a CEOM square with a side of 5 cm. Determine the perimeter of the square.
Where is the calculation of the perimeter of a rectangle used?
1. A piece of land is given, it needs to be surrounded by a fence. How long will the fence be?
In this task, it is necessary to accurately calculate the perimeter of the site so as not to buy extra material for building a fence.
2. Parents decided to make repairs in the children's room. You need to know the perimeter of the room and its area in order to correctly calculate the number of wallpapers.
Determine the length and width of the room you live in. Determine the perimeter of your room.
What is the area of a rectangle?
Square- This is a numerical characteristic of the figure. The area is measured in square units of length: cm 2, m 2, dm 2, etc. (centimeter squared, meter squared, decimeter squared, etc.)In calculations, it is denoted by the Latin letter S.
To find the area of a rectangle, multiply the length of the rectangle by its width. 
The area of the rectangle is calculated by multiplying the length of AK by the width of KM. Let's write this as a formula.
S AKMO=AK*KM
Example.
What is the area of rectangle AKMO if its sides are 7 cm and 2 cm?
S AKMO \u003d AK * KM \u003d 7 cm * 2 cm \u003d 14 cm 2.
Answer: 14 cm 2.
The formula for calculating the area of a square
The area of a square can be determined by multiplying the side by itself.Example. 
In this example, the area of the square is calculated by multiplying side AB by width BC, but since they are equal, side AB is multiplied by AB.
S ABCO = AB * BC = AB * AB
Example.
Find the area of the square AKMO with a side of 8 cm.
S AKMO = AK * KM = 8 cm * 8 cm = 64 cm 2
Answer: 64 cm 2.
Problems to find the area of a rectangle and a square
1. A rectangle with sides of 20 mm and 60 mm is given. Calculate its area. Write your answer in square centimeters.2. A suburban area was bought with a size of 20 m by 30 m. Determine the area suburban area Write your answer in square centimeters.
