Perimeter and area of a rectangle. Perimeter and area of a rectangle How to find the perimeter and area of a rectangle formula
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 be done without the perimeter formula and the skills to calculate it.
Basic Concepts
A rectangle is a quadrilateral in which all the angles are right and the opposite sides are equal and parallel in pairs. In our life, many figures have the shape of a rectangle, for example, the surface of a table, a notebook, etc.
Let's look at an example: A fence must be erected along the boundaries of the land plot. In order to find out the length of each side, you need to measure them.
Rice. 1. A plot of land in the shape of a rectangle.
The plot of land has sides with lengths of 2 m, 4 m, 2 m, 4 m. Therefore, to find out the total length of the fence, you need to add up the lengths of all sides:
2+2+4+4= 2·2+4·2 =(2+4)·2 =12 m.
It is this quantity that is generally called the perimeter. Thus, to find the perimeter, you need to add up all the sides of the figure. The letter P is used to denote the perimeter.
To calculate the perimeter of a rectangular figure, you do not need to divide it into rectangles; you only need to measure 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 is 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 measurement system.
Formula for the perimeter of a figure
If we take into account the fact that the 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 the figure, you can use to express the other side in terms of 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 sides of a figure, the semi-perimeter is the sum of only one length and width. To get the perimeter, you need to multiply the semi-perimeter by 2.
Area and perimeter are 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 learned how to find the perimeter of a rectangle. We also got acquainted with the formula for calculating it. This topic can be encountered not only when solving mathematical problems, but also in real life.
Test on the topic
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Perimeter is the sum of the lengths of all sides of the polygon.
- To calculate the perimeter of geometric figures, special formulas are used, where the perimeter is denoted by the letter “P”. It is recommended to write the name of the figure in small letters under the sign “P” so that you know whose perimeter you are finding.
- The perimeter is measured in units of length: mm, cm, m, km, etc.
Distinctive features of a rectangle
- A rectangle is a quadrilateral.
- All parallel sides are equal
- All angles = 90º.
- For example, in everyday life, a rectangle can be found in the form of a book, monitor, table cover or door.
How to calculate the perimeter of a rectangle
There are 2 ways to find it:
- 1 way. Add up all sides. P = a + a + b + b
- Method 2. Add the width and length and multiply by 2. P = (a + b) 2. OR P = 2 a + 2 b. The sides of a rectangle that lie opposite each other (opposite) are called length and width.
"a"- the length of a rectangle, the longer pair of its sides.
"b"- the width of the rectangle, the shorter pair of its sides.
An example of a problem to calculate the perimeter of a rectangle:
Calculate the perimeter of the rectangle, its width is 3 cm, and its length is 6.
Remember the formulas for calculating the perimeter of a rectangle!
Semiperimeter is the sum of one length and one width .
- Semi-perimeter of a rectangle - when you perform the first action in brackets - (a+b).
- To obtain a perimeter from a semi-perimeter, you need to increase it by 2 times, i.e. multiply by 2.
How to find the area of a rectangle
Rectangle area formula S= a*b
If the length of one side and the length of the diagonal are known in the condition, then the area can be found using the Pythagorean theorem in such problems; it allows you to find the length of a side of a right triangle if the lengths of the other two sides are known.
- : a 2 + b 2 = c 2, where a and b are the sides of the triangle, and c is the hypotenuse, the longest side.
Remember!
- All squares are rectangles, but not all rectangles are squares. Because:
- Rectangle is a quadrilateral with all right angles.
- Square- a rectangle with all sides equal.
- If you find the area, the answer will always be in square units (mm 2, cm 2, m 2, km 2, etc.)
Among the inexhaustible variety of geometric shapes, there are those that are most applicable in our lives, for example, a parallelogram, a circle, an oval, etc. Geometric shapes are everywhere, in connection with this there is often a need to determine their numerical characteristics: area, perimeter, volume.
A rectangle has many distinctive features, on the basis of which rules for calculating its various numerical characteristics have been developed. So, a rectangle:- it is a flat geometric figure;
- it is a quadrilateral;
- This is a figure in which opposite sides are equal and parallel, all angles are right, i.e. at 90°.
- there is a rectangle ABCD;
- sides AB and CD are 5 cm;
- sides BC and AD are 7 cm.
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 right angles. This means that opposite sides are equal to each other.
Square is a rectangle with equal sides and equal angles. It is called a regular quadrilateral.
Quadrangles, including rectangles and squares, are designated 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 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 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 rectangle ABCD is denoted as P ABCD, where A, B, C, D are the vertices of the rectangle.
Let's write down the formula for the perimeter of a quadrilateral ABCD:
P ABCD = AB + BC + CD + AD = 2 * AB + 2 * BC = 2 * (AB + BC)
Example.
Given a rectangle ABCD with sides: AB=CD=5 cm and AD=BC=3 cm.
Let's define P ABCD.
Solution:
1. Let's draw a rectangle ABCD with the original data. 
2. Let’s write a formula to calculate the perimeter of a given rectangle:
P ABCD = 2 * (AB + BC)
P ABCD = 2 * (5 cm + 3 cm) = 2 * 8 cm = 16 cm
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 determine 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. Let us determine the perimeter of the square.
Solution.
1. Let's draw a square ABCD with the original data.
2. Let us 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 * 6 cm = 24 cm
Answer: P ABCD = 24 cm.
Problems to find 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 SEOM 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 plot of land has been 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 excess material for building a fence.
2. Parents decided to renovate the children's room. You need to know the perimeter of the room and its area in order to correctly calculate the amount of wallpaper.
Determine the length and width of the room in which you live. Determine the perimeter of your room.
What is the area of a rectangle?
Square is a numerical characteristic of a figure. 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 AC by the width of the CM. Let's write this down 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 = 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. 
In this example, the area of the square is calculated by multiplying the side AB by the width BC, but since they are equal, the result is multiplying the side AB by AB.
S ABCO = AB * BC = AB * AB
Example.
Determine the area of a 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 square
1. Given a rectangle with sides 20 mm and 60 mm. Calculate its area. Write your answer in square centimeters.2. A dacha plot measuring 20 m by 30 m was purchased. Determine the area of the dacha plot and write the answer in square centimeters.
Sections: Primary School
Class: 3
Lesson topic: formulas for the perimeter and area of a rectangle.
Lesson type: lesson introducing new knowledge.
Objective of the lesson: construct a formula for finding the side of a rectangle along its perimeter and the other side.
1) form an idea of the formula as an equality that establishes a relationship between quantities. To teach, in the simplest cases, to express the relationship between quantities using formulas. Practice oral and written calculation skills.
2) Develop the ability to analyze, compare, generalize.
3) Develop communication skills and speech culture.
Equipment: form with tasks
During the classes
1. Self-determination for activity.
Mathematics has arrived
Take your seats.
Find something useful to do for your head!
So as not to yawn from idleness,
It’s useful to “rack your brains”!
How do you understand the phraseological phrase “puzzle”?
2. Updating knowledge.
1) What do the records have in common?
2 x = 480
Y – 56 = 64
A=S:b
d: 5=12
S = a b
540: z = 18
P = (a+b) 2
(These are equalities containing variables.)
2) What groups can they be divided into?
(Equations and formulas.)
3) What is an equation called? (Equality with the variable whose value needs to be found.)
4) Find the roots of the equations and write them separated by commas in your notebook. (240, 120, 60, 30.)
5) What interesting things did you notice? (All numbers are round, each subsequent one is reduced by 2 times.)
6) What number is next? (15)
7) Write it down, mentally remove the commas and read the resulting number. (240 120 603 015.)
8) Look at the equalities in the second column. What does the first formula show? Second? And the third?
9) How do formulas differ from equations? (In equations, letters represent certain numbers, and in formulas, the values of quantities; formulas are valid for all letter values, and equations are valid only for roots)
10) What are formulas for?
11) What word is the word “formula” similar to? (The word “formula” is similar to the word “form”. A mold for sand helps to make pies out of it, and formulas help solve problems by specifying the form of relationships between quantities)
12) Try to formulate a definition of the formula.
(A formula is a correct equality that establishes the relationship between quantities)
3. Statement of the educational task.
Using these formulas, solve problems No. 1, No. 2, No. 3 from the problem form. You will work in pairs.
1) Find the area of a rectangle with sides 30 cm and 80 cm.
2) Find the side of a rectangle whose area is 1800 square meters. cm, and the second side is 20 cm.
3) The width of the rectangle is 8 cm. What is the length if the perimeter is 40 cm?
4) The length of the rectangle is 3 m and the width is 2 dm. What is the perimeter?
5) The width of the rectangle is 6 cm. What is the length if the perimeter is 44 cm?
6) The length of the rectangle is 5 cm and the width is 10 mm shorter. What is its perimeter?
Checking the solution.
What formula helped solve the first problem? Second?(S = a b), (a =S: b)
Why couldn't they solve the third problem? (The required formula is not in the list of formulas we have studied)
So what will we do in class? (We will derive a formula for finding the side of a rectangle through the perimeter and the other side)
The topic of our lesson: “Formulas for the perimeter and area of a rectangle.”
4. “Discovery” of new knowledge by children.
1) Where do we start? (Let's build a drawing and introduce notations)
Children can derive a formula based on logical reasoning based on a drawing. The sum of the length and width is half the perimeter, and to find one of the sides, you need to subtract the other side from this half: a = P: 2 - b
Second way.
2) What does this formula resemble: P= (a+b) · 2? (The equation)
3) What is this equation? (This is a compound equation)
4) What is the sum of a and b? (First factor)
5) How to find an unknown factor? (a+b=P:2)
6) What is unknown to us now? (term)
7) How to find an unknown term? (a=P:2-b)
So, we have derived the formula for finding the length of a rectangle. What will the formula for finding the width of a rectangle look like? (b=P:2-a)
What is the formula? (A formula is a correct equality that establishes the relationship between quantities)
Read the resulting formula. (The length of a side of a rectangle is equal to the difference between half the perimeter and the length of the other side)
Now, using the new formula, let's solve the problem that you couldn't solve.
b=P: 2-a=40:2-8=12(cm)
5. Physical exercise.
The sun came into the classroom
It calls us all to rest.
One two three four five
We need to sit down and stand up.
Stretch your arms wider
One two three four five
Bend over - three or four,
And jump on the spot.
On the toe, then on the heel,
We all do exercises.
6. Primary consolidation in external speech.
1)Look at the remaining tasks. Which one can we solve using the newly derived formula? (No. 4)
b = P: 2 – a = 44: 2- 6 = 16 (cm)
Is there another way to solve this problem? (Substitute known quantities into the formula)
P= (a+b) 2
44= (6+b) 2
(6+b) 2=44
6 + b = 44: 2
6+b=22
b=22-6
b=16
Answer: the length of the rectangle is 16 cm.
7. Independent work with self-test according to options:
Work from the textbook: Peterson L. G. Mathematics. 3rd grade. Part 2. – M.: Yuventa Publishing House, 2005. – 96 p.: ill. :
1 option No. 4 (p. 86)
Option 2 No. 6 (p. 87)
On the desk:
3 m =30 dm
P=(30+2) 2=64 (dm)
10 mm =1 cm
5-1=4(cm)
P=(5+4) 2=18(cm)
8. Inclusion in the knowledge system and repetition.
Solving equations from No. 7(a, e) based on the previously derived algorithm.
9. Reflection of activity.
What is the purpose of our lesson?
Have we achieved our goal?
How do you evaluate your work?
10. Homework.
Learn the formulas from the reference notes in the textbook on page 86 and solve problems from No. 3, page 87.
Literature
1. Peterson L.G. Mathematics. 3rd grade. Part 2. – M.: Yuventa Publishing House, 2005. – 96 p.: ill.
