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 the area and perimeter can be useful in Everyday life... In everyday life, these values are used when it is necessary to paint something, plant or process a garden, glue wallpaper in the room, etc.
Perimeter
Most often, you need to know the perimeter of polygons or triangles. To determine this value, you just need 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 = 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.
Circle
The perimeter of a circle is commonly referred to as the circumference. To find out this value, you must use the formula: L = π * D = 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 both sides of one figure and the other are equal. Since the square and the rhombus have equal sides, they (sides) can be designated by one letter "a". It turns out that the perimeter of the square and the rhombus is:
- P = a + a + a + a or P = 4a
Rectangle, parallelogram
A rectangle and a parallelogram have the same opposite sides, so they can be designated by two different letters "a" and "b". The formula looks like this:
- P = a + b + a + b = 2a + 2b. The two can be taken out of the brackets, and you get the following formula: P = 2 (a + b)
Trapezoid
In a trapezoid, all sides are different, so they are designated 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
An area is that part of a figure that is enclosed within its outline.
Rectangle
To calculate the area of a rectangle, multiply the value of one side (length) by the value of the other (width). If the length and width values are designated 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 of the square:
- S = a * a = a 2
Rhombus
The formula for finding the area of a rhombus has a slightly different form: S = 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 = a 2 * sin α, while a is the side of the figure, and the angle α is the angle between the sides;
- S = 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.
Circle
The area of the circle is also easy to recognize. To do this, you can use the formula:
- S = πR 2, where R is the radius.
Trapezoid
To calculate the area of a trapezoid, you can use this formula:
- S = 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 = 1/2 * a * b sin α (where a, b are the sides of the triangle, and α is the angle between them);
- S = 1/2 a * h (where a is the base of the triangle, h is the height lowered to it);
- S = abc / 4R (where a, b, c are the sides of the triangle, and R is the radius of the circumscribed circle);
- S = p * r (where p is a semi-perimeter, r is the radius of the inscribed circle);
- S = √ (p * (p-a) * (p-b) * (p-c)) (where p is the semiperimeter, a, b, c are the sides of the triangle).
Parallelogram
To calculate the area of a given figure, you must substitute the values in one of the formulas:
- S = a * b * sin α (where a, b are the bases of the parallelogram, α is the angle between the sides);
- S = 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).
Lesson and presentation on the topic: "Perimeter and area of a rectangle"
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What are rectangle and square
Rectangle Is a quadrilateral with all angles right. This means that the opposite sides are equal to each other.
Square Is a rectangle with equal sides and corners. It is called a regular quadrilateral.
Quadrangles, including rectangles and squares, are denoted by 4 letters - vertices. To designate the vertices, Latin letters are used: A, B, C, D...
Example. 
It reads like this: quadrilateral ABCD; square EFGH.
What is the perimeter of a rectangle? Perimeter calculation formula
Perimeter of a rectangle Is the sum of the lengths of all sides of the rectangle or the sum of the length and width times 2.The perimeter is denoted by a 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 the 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 the quadrangle ABCD:
P ABCD = AB + BC + CD + AD = 2 * AB + 2 * BC = 2 * (AB + BC)
Example.
A rectangle ABCD with sides is given: AB = СD = 5 cm and AD = BC = 3 cm.
Let us define P ABCD.
Solution:
1. Let's draw a rectangle ABCD with the original data. 
2. Let's write a formula for calculating the perimeter of a given rectangle:
P ABCD = 2 * (AB + BC)
P ABCD = 2 * (5cm + 3cm) = 2 * 8cm = 16cm
Answer: P ABCD = 16 cm.
Formula for calculating the perimeter of a square
We have a formula for determining the perimeter of a rectangle.P ABCD = 2 * (AB + BC)
Let's use it to define the perimeter of the square. Considering that all sides of the square are equal, we get:
P ABCD = 4 * AB
Example.
A square ABCD with a side equal to 6 cm is given. Let us determine the perimeter of the square.
Solution.
1. Let's draw a square ABCD with the original data.
2. Recall the formula for calculating the perimeter of a square:
P ABCD = 4 * AB
3. Let's substitute our data into the formula:
P ABCD = 4 * 6cm = 24cm
Answer: P ABCD = 24 cm.
Tasks 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 square СEOM with a side of 5 cm. Determine the perimeter of the square.
Where is the calculation of the perimeter of a rectangle used?
1. Given a piece of land, 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. The 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 Is a numerical characteristic of a 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 a Latin letter S.
To determine 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 the AK by the width of the CM. Let's write it down as a formula.
S AKMO = AK * KM
Example.
What is the area of an AKMO rectangle if its sides are 7 cm and 2 cm?
S AKMO = AK * KM = 7 cm * 2 cm = 14 cm 2.
Answer: 14 cm 2.
Formula for calculating the area of a square
The area of a square can be determined by multiplying the side by itself.Example. 
V this example the area of the square is calculated by multiplying side AB by the width of BC, but since they are equal, it multiplies side AB by AB.
S ABCO = AB * BC = AB * AB
Example.
Determine the area of an AKMO square with a side of 8 cm.
S AKMO = AK * KM = 8 cm * 8 cm = 64 cm 2
Answer: 64 cm 2.
Tasks for finding the area of a rectangle and 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 summer cottage plot measuring 20 m by 30 m was purchased. Determine the area suburban area, write your answer in square centimeters.
Definition.
Rectangle- this is a quadrangle in which two opposite sides are equal and all four corners are the same.The rectangles differ from each other only in the ratio of the long side to the short side, but all four corners are straight, that is, 90 degrees.
The long side of the rectangle is called the length of the rectangle, and the short one - width of the rectangle.
The sides of the rectangle are also its heights.
Basic properties of a rectangle
The rectangle can be a parallelogram, square, or rhombus.
1. Opposite sides of a rectangle have the same length, that is, they are equal:
AB = CD, BC = AD
2. Opposite sides of the rectangle are parallel:
3. Adjacent sides of the rectangle are always perpendicular:
AB ┴ BC, BC ┴ CD, CD ┴ AD, AD ┴ AB
4. All four corners of the rectangle are straight:
∠ABC = ∠BCD = ∠CDA = ∠DAB = 90 °
5. The sum of the angles of the rectangle is 360 degrees:
∠ABC + ∠BCD + ∠CDA + ∠DAB = 360 °
6. The diagonals of the rectangle are of the same length:
7. The sum of the squares of the diagonal of the rectangle is equal to the sum of the squares of the sides:
2d 2 = 2a 2 + 2b 2
8. Each diagonal of the rectangle divides the rectangle into two identical shapes, namely, right-angled triangles.
9. The diagonals of the rectangle intersect and are halved at the intersection:
| AO = BO = CO = DO = | d | ||
| 2 |
10. The point of intersection of the diagonals is called the center of the rectangle and is also the center of the circumscribed circle
11. The diagonal of a rectangle is the diameter of the circumscribed circle
12. Around a rectangle, you can always describe a circle, since the sum of opposite angles is 180 degrees:
∠ABC = ∠CDA = 180 ° ∠BCD = ∠DAB = 180 °
13. In a rectangle whose length is not equal to the width, it is impossible to inscribe a circle, since the sums of opposite sides are not equal to each other (you can inscribe a circle only in special case rectangle - square).
Sides of a rectangle
Definition.
The length of the rectangle is the length of the longer pair of its sides. Width of the rectangle is the length of the shorter pair of its sides.Formulas for determining the lengths of the sides of a rectangle
1. Formula of the side of a rectangle (length and width of the rectangle) through the diagonal and the other side:
a = √ d 2 - b 2
b = √ d 2 - a 2
2. Formula of the side of a rectangle (length and width of the rectangle) through the area and the other side:
| b = d cos | β |
| 2 |
Diagonal of a rectangle
Definition.
Diagonal rectangle any segment connecting two vertices of opposite corners of a rectangle is called.Formulas for determining the length of the diagonal of a rectangle
1. The formula for the diagonal of a rectangle through the two sides of a rectangle (through the Pythagorean theorem):
d = √ a 2 + b 2
2. Formula of the diagonal of a rectangle in terms of the area and any side:
4. Formula of the diagonal of a rectangle in terms of the radius of the circumscribed circle:
d = 2R
5. Formula of the diagonal of a rectangle through the diameter of the circumscribed circle:
d = D about
6. Formula of the diagonal of a rectangle in terms of the sine of the angle adjacent to the diagonal, and the length of the side opposite to this angle:
8. Formula of the diagonal of a rectangle in terms of sine acute angle between the diagonals and the area of the rectangle
d = √2S: sin β
Perimeter of a rectangle
Definition.
Perimeter of a rectangle called the sum of the lengths of all sides of the rectangle.Formulas for determining the length of the perimeter of a rectangle
1. Formula for the perimeter of a rectangle through two sides of the rectangle:
P = 2a + 2b
P = 2 (a + b)
2. Formula for the perimeter of a rectangle in terms of the area and any side:
| P = | 2S + 2a 2 | = | 2S + 2b 2 |
| a | b |
3. Formula for the perimeter of a rectangle through the diagonal and any side:
P = 2 (a + √ d 2 - a 2) = 2 (b + √ d 2 - b 2)
4. Formula for the perimeter of a rectangle in terms of the radius of the circumscribed circle and any side:
P = 2 (a + √4R 2 - a 2) = 2 (b + √4R 2 - b 2)
5. Formula for the perimeter of a rectangle in terms of the diameter of the circumscribed circle and any side:
P = 2 (a + √D o 2 - a 2) = 2 (b + √D o 2 - b 2)
Rectangle area
Definition.
By the area of the rectangle is called the space bounded by the sides of the rectangle, that is, within the perimeter of the rectangle.Formulas for determining the area of a rectangle
1. Formula for the area of a rectangle in two sides:
S = a b
2. Formula for the area of a rectangle in terms of the perimeter and any side:
5. Formula of the area of a rectangle in terms of the radius of the circumscribed circle and any side:
S = a √4R 2 - a 2= b √4R 2 - b 2
6. Formula of the area of a rectangle in terms of the diameter of the circumscribed circle and any side:
S = a √D o 2 - a 2= b √D o 2 - b 2
Circle circumscribed around a rectangle
Definition.
Circled around a rectangle is called a circle passing through the four vertices of a rectangle, the center of which lies at the intersection of the diagonals of the rectangle.Formulas for determining the radius of a circle circumscribed around a rectangle
1. Formula for the radius of a circle circumscribed around a rectangle through two sides:
A rectangle is a special case of a quadrangle. This means that the rectangle has four sides. Its opposite sides are equal: for example, if one of its sides is 10 cm, then the opposite side will also be 10 cm. A special case of a rectangle is a square. A square is a rectangle with all sides equal. To calculate the area of a square, you can use the same algorithm as to calculate the area of a rectangle.
How to find out the area of a rectangle on two sides
In order to find the area of a rectangle, you need to multiply its length by width: Area = Length × Width. In the case below: Area = AB × BC.
How to find out the area of a rectangle by the side and length of the diagonal
In some problems it is necessary to find the area of a rectangle using the length of the diagonal and one of the sides. The diagonal of the rectangle divides it into two equal right triangle... Therefore, it is possible to determine the second side of the rectangle using the Pythagorean theorem. After that, the task is reduced to the previous point.
How to find out the area of a rectangle along the perimeter and side
The perimeter of a rectangle is the sum of all its sides. If you know the perimeter of the rectangle and one side (for example, the width), you can calculate the area of the rectangle using the following formula:
Area = (Perimeter × Width - Width ^ 2) / 2.
Area of a rectangle through the sine of an acute angle between diagonals and the length of a diagonal
The diagonals in the rectangle are equal, so to calculate the area based on the length of the diagonal and the sine of the acute angle between them, use the following formula: Area = Diagonal ^ 2 × sin (acute angle between diagonals) / 2.
When solving, it is necessary to take into account that to solve 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 pick up endless number of options the sizes of the sides of the rectangle, the perimeter of which will be equal to the specified 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 = 1 * 9 = 9 cm 2
S 2 = 2 * 8 = 16 cm 2
S 3 = 3 * 7 = 21 cm 2
As you can see, there are an infinite number of options for the area of a figure for a given perimeter.
Thus, in order to calculate the area of a rectangle from its perimeter, it is necessary to know either the aspect ratio or the length of one of them. The only figure that has an unambiguous dependence of its area on the perimeter is a circle. For circle only and a solution is possible.
In this tutorial:
- Problem 4. Changing the length of the sides while maintaining the area of the rectangle
Problem 1. Find the sides of a rectangle from the area
The perimeter of the 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.Solution.
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
2x 2 + 2y 2 = 260
x + y = 16
x = 16-y
2 (16-y) 2 + 2y 2 = 260
2 (256-32y + y 2) + 2y 2 = 260
512-64y + 4y 2 -260 = 0
4y 2 -64y + 252 = 0
D = 4096-16x252 = 64
x 1 = 9
x 2 = 7
Now let's take into account that based on the fact that x + y = 16 (see above) for x = 9, then y = 7 and vice versa, if x = 7, then y = 9
Answer: The sides of the rectangle are 7 and 9 centimeters
Problem 2. Find the sides of a rectangle from the perimeter
The perimeter of the rectangle is 26 cm, and the sum of the areas of the squares built on its two adjacent sides is 89 sq. see Find the sides of the rectangle.
Solution.
Let's designate 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 (squares, respectively, two and these are squares of width and height, since the sides are adjacent) will be equal
x 2 + y 2 = 89
We solve the resulting system of equations. From the first equation we deduce that
x + y = 13
y = 13-y
Now we substitute into the second equation, replacing x with its equivalent.
(13-y) 2 + y 2 = 89
169-26y + y 2 + y 2 -89 = 0
2y 2 -26y + 80 = 0
We solve the resulting quadratic equation.
D = 676-640 = 36
x 1 = 5
x 2 = 8
Now let's take into account that based on the fact that x + y = 13 (see above) for x = 5, then y = 8 and vice versa, if x = 8, then y = 5
Answer: 5 and 8 cm
Problem 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.
Solution.
Let's denote the sides of the rectangle through the proportionality coefficient x.
Whence the length of one side will be 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 cm 2
Problem 4... Change the length of the sides while maintaining the area of the rectangle
The length of the rectangle is increased by 25%. By what percentage should the width be reduced so that its area does not change?Solution.
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 = 1.25ab
Thus, in order to return the area of the rectangle to initial value, then
S 2 = S / 1.25
S 2 = 1.25ab / 1.25
Insofar as new size but you cannot change, then
S 2 = (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: the width needs to be reduced by 20%.