Calculator for calculating the perimeter and area of ​​geometric shapes. Rectangle

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 by the simplest way of 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).

Area 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:

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 hang 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.

A 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.

Etc. 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 ​​a dodecagon can be a difficult task. Use our catalog of calculators to solve school assignments or everyday problems.

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.

The task : 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.

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 cultivate 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.

Area

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).

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 .

Solution:
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.

Solution.
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?

Area- 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 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.
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 of ​​\u200b\u200bthe summer cottage, write down the answer in square centimeters.