Finding the perimeter and area of a rectangle. How to find the perimeter and area of a rectangle. Where is the calculation of the perimeter of a rectangle used?
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
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. Perimeter of a rectangle is equal to the sum 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 in shape and you need to fence the site, 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.
Let's remember school course mathematics. 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).
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:
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).
Among the inexhaustible variety of geometric shapes, there are those that are most applicable in our life, for example, a parallelogram, a circle, an oval, etc. Geometric figures everywhere, in connection with this, it often becomes necessary to determine their numerical characteristics: area, perimeter, volume.
The rectangle has many distinctive features, on the basis of which the rules for calculating its various numerical characteristics have been developed. So the 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. by 90°.
- there is a rectangle ABCD;
- sides AB and CD are 5 cm;
- sides BC and AD are 7 cm.
The ability to find the perimeter of a rectangle is very important for solving many geometric problems. Below is a detailed instruction on finding 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 the 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.
Since we know the side of the square, we can find its perimeter.
Add up all sides: 5 + 5 + 5 + 5 = 20.
Answer: P = 20 cm.
Multiply the side of the square by 4 (because everyone is equal): 4x5 = 20.
Answer: P = 20 cm.
How to Find the Perimeter of a Rectangle - Online Resources
While the steps above are easy to understand and master, there are several online calculators that can help you calculate the perimeters (area, volume) of different shapes. Just type in the required values and the mini-program will calculate the perimeter of the shape you need. Below is a short list.
Perimeter is a geometric term that is often found in problems. To understand what a perimeter is, you should draw an arbitrary polygon and arm yourself with a ruler. Translated from Greek this term means "measuring around".
How to calculate the perimeter
The perimeter is indicated by the Latin letter P. It can be measured in centimeters, millimeters, meters or decimeters. To find out the perimeter, you should measure the length of all sides of the polygon. The resulting values must be added. The final sum will be the answer to the question: "What is the perimeter of the polygon."
The perimeter is the length of lines that bound a closed figure (square, rectangle, triangle, etc.).
For example, in front of you is a polygon with sides of 10, 12, 13 and 11 cm. Add the above numbers (10 + 12 + 13 + 11) and get the sum 46. This is the perimeter of the polygon.
For the convenience of calculating the perimeter in geometry, there are a number of formulas. Each formula corresponds to a specific figure.
Perimeter and area of a square
This is the sum of its four sides. As we know, all sides of a square are the same size. Therefore, we can find out the perimeter of a square by multiplying its side length by four:
P=a+a+a+a
For example, we have a square with a side of 10 cm.
Answer: 40 cm
P= 10+10+10+10
P=40
Answer: 40 cm
To understand what perimeter and area are, it should be understood that the perimeter calculates the length of the contour of the figure, and the area calculates the size of its entire surface.
To find out the area of a square, you need to use a simple formula:
S is the area, and is the side of the square.
For example, in the problem it is indicated that the length of the side of the square is 10 cm.
S=100cm 2
Answer: 100 cm 2
Perimeter and area of a rectangle
Sides of a rectangle that are opposite each other and have the same length are called opposite sides. This is the length and width, they are conventionally denoted by the Latin letters a and b. The formula for calculating the perimeter of a rectangle looks like this:
P=(a+b)*2
Using this formula, we first find the sum of the width and length and then multiply it by two.
For example, we have a rectangle with a length of 6 cm and a width of 2 cm.
P= (6+2) * 2
P= 16
Answer: 16 cm
To find the area of a rectangle, multiply the length by the width. The formula looks like this:
For example, in the conditions of the problem it is said that the rectangle has a length of 5 cm and a width of 2 cm. Change the letters a and b to the indicated numbers.
S= 5*2
S\u003d 10cm 2
Answer: 10 cm 2
Circle perimeter (circumference)
Each circle has a center. The distance from the center of the circle to any point on the circle is called the radius of the circle. Often students confuse the concepts of "circle" and "circumference" and try to determine the area of a circle. This is a serious mistake. It is necessary to separate the concepts of "circle" and "circumference" in the head. A circle does not and cannot have an area, it has only a length.
To find the perimeter of a circle, calculate the circumference of its circumference. There is a formula for finding the circumference of a circle:
L = 2πr
L- circumference
π is the number "pi", a mathematical constant. It is equal to the ratio of the circumference of a circle to the length of its diameter. The ancient name for the number "pi" is the Ludolf number. This number is irrational, its decimal representation never ends after the dot.
π = 3.141 592 653 589 793 238 462 643 383 279 502
For the convenience of calculations, the value 3.14 is usually used
R is the radius of the circle
D– Circle diameter
So, to determine the perimeter of a circle, you need to find the product of the radius and 2π. If the problem specifies a diameter, then
For example, we have a circle with a radius of 3 cm in front of us. Let's find its perimeter.
L= 2*3,14*3
L=6 π
L=6*3.14
L= 18.84 cm
Pto= 18.84 cm
Answer: 18.84 cm
Difference between perimeter and area
The area is the size of the surface of the figure, and the perimeter is the sum of its boundaries.
The area is always measured in square units (cm 2, m 2, mm 2). The perimeter is measured in units of length - in centimeters, millimeters, meters, decimeters.
