How to get perimeter from area. Perimeter and area of a rectangle
Geometry comprehends the properties and combinations of two-dimensional and spatial figures. The numerical values characterizing such structures are square and perimeter, the calculation of which is carried out using famous formulas or is expressed one through the other.
Instructions
1. Rectangle.Task: calculate square a rectangle, if we know that its perimeter is 40, and its length b is 1.5 times greater than its width a.
2. Solution: Use the famous perimeter formula, it is equal to the sum of all sides of the figure. In this case P = 2 a + 2 b. From the initial data of the problem, you know that b = 1.5 a, therefore, P = 2 a + 2 1.5 a = 5 a, whence a = 8. Find the length b = 1.5 8 = 12.
3. Write down the formula for the area of a rectangle: S = a b, Substitute the known quantities: S = 8 * 12 = 96.
4. Square.Task: discover square square if the perimeter is 36.
5. Solution: A square is a special case of a rectangle, where all sides are equal, therefore, its perimeter is 4 a, whence a = 8. Determine the area of the square using the formula S = a? = 64.
6. Triangle.Problem: given an arbitrary triangle ABC, the perimeter of which is 29. Find out the value of its area if it is known that the height BH, lowered onto side AC, divides it into segments with lengths 3 and 4 cm.
7. Solution: First, remember the area formula for a triangle: S = 1/2 c h, where c is the base and h is the height of the figure. In our case, the base will be the side AC, which is known from the condition of the problem: AC = 3+4 = 7, it remains to find the height BH.
8. The altitude is a perpendicular drawn to the side from the opposite vertex, therefore, it divides triangle ABC into two right triangles. Knowing this quality, look at the triangle ABH. Remember the Pythagorean formula, according to which: AB? = BH? +AH? = BH? + 9 ? AB = ?(h? + 9). In the triangle BHC, according to the same thesis, write: BC? = BH? +HC? = BH? + 16 ? BC = ?(h? + 16).
9. Apply the perimeter formula: P = AB + BC + AC Substitute the values expressed in terms of height: P = 29 = ?(h? + 9) + ?(h? + 16) + 7.
10. Solve the equation:?(h? + 9) + ?(h? + 16) = 22? [replacement t? = h? + 9]:?(t? + 7) = 22 – t, square both sides of the equation:t? + 7 = 484 – 44 t + t? ? t?10.84h? + 9 = 117.5? h? 10.42
11. Discover square triangle ABC:S = 1/2 7 10.42 = 36.47.
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 to you a catalog of online calculators that allow you to calculate the parameters of the most popular plane figures. Let's look at them.
Circle
Special cases
A quadrilateral with equal sides. A parallelogram becomes a rhombus when its diagonals intersect at an angle of 90 degrees and are bisectors of their angles.
This 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.
This 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, which is characterized by maximum symmetry.
Polygon
A regular polygon is a convex figure on a plane that has equal sides and equal angles. Depending on the number of sides, polygons have their own names:
- - Pentagon;
- - hexagon;
- eight - octagon;
- twelve is a dodecagon.
And so on. 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 only regular polygons. It uses general formulas for all valid polygons. To calculate the perimeter, use the formula:
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).
By 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. Thus, the building of the US Department of Defense - the Pentagon - has the shape of a pentagon; a hexagon - a honeycomb 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 calculations.
Painting the fence
Painting surfaces and calculating paint are some of the most obvious everyday tasks that require minimal mathematical calculations. If we need to paint a fence whose height is 1.5 meters and length is 20 meters, then how many cans of paint will be needed? To do this, you need to find out the total area of the 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 a calculator to calculate the area of a 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 square meters. Then we will need 60 × 0.13 = 7.8 kilograms of paint or three standard 2.8 kilogram cans.
Fringe trim
Tailoring is another industry that requires extensive geometric knowledge. Suppose we need to trim a scarf with fringe, 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 an online calculator comes in handy. Let's enter this cell data and get the answer:
Thus, we will need 4 m of fringe to finish the scarf.
Conclusion
Flat figures make up the real world around us. We often wondered at school whether geometry would 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 issues.
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 equal to 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 is exactly the same as (2 + 8) * 2 = 20 cm.
As you can see, we can select endless number of options the dimensions 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 example given - 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, you must 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 its perimeter is a circle. Only for circle and a solution is possible.
In this lesson:
- 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 conditions of the problem, the sum of the areas of the squares constructed on each of its sides (four squares, respectively) 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) at 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 square meters. cm. Find the sides of the rectangle.
Solution.
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 squares of width and height, since the sides are adjacent) will be equal to
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 perform a substitution in 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) at 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 its sides are proportional as 2 to 3.
Solution.
Let us denote the sides of the rectangle by the proportionality coefficient x.
Hence 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 cm 2
Problem 4. Changing 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 equal to
S2 = 1.25ab
Thus, in order to return the area of the rectangle to the initial value, then
S2 = S/1.25
S2 = 1.25ab / 1.25
Since the new size a cannot be changed, 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: width should be reduced by 20%.
How to calculate the area of a figure knowing its perimeter? and got the best answer
Answer from Yeomen Arkadyevich[guru]
In Compass 3D, draw a plan and automatically calculate the area. The area of an arbitrary polygon cannot be calculated along the perimeter. You still have to break it down into separate figures.
If you have any questions, write to the agent.
Answer from Yamis Sh[newbie]
..
Answer from Kiss(RUSS for all) ki (I)[guru]
1.select center
2.measure the distance from the center to the corners
3.measure the sides of your polygon
4.calculate the perimeters of the resulting N triangles
5.calculate the areas of all triangles using Heron’s formula through the semiperimeter.
6.sum all areas
7.choose my answer as the best.
8.all
Answer from Semrid[guru]
try dividing the perimeter by 4 and then multiplying the results by each other
Answer from ScrAll[guru]
Cut it out of paper and weigh it.
Or you break it into triangles.
Half the base to the height...
Answer from Alexey Zaitsev[guru]
It is easier and more error-free to draw a sketch - a top view with dimensions. Then, using this sketch, divide the area into rectangles, calculate and sum their areas
Answer from Maria Kempel[active]
unreal
Answer from Nemo[guru]
Unreal. Along the perimeter, the area of only REGULAR figures is calculated. I recommend the piecewise method
Answer from Jon[guru]
it is best to break a complex figure into several simple ones, and calculate the area separately, then add
Answer from Lavavoth[guru]
Unreal.. . It’s better to post the floor plan of the hall, there are other ways of counting, but you need to see the plan.
Answer from 3 answers[guru]
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