Как найти периметр квадрата когда известна площадь

Квадрат представляет собой правильный четырехугольник (или ромб), в котором все углы являются прямыми, а стороны равны между собой. Как и у любого иного правильного многоугольника, у квадрата можно высчитать периметр и площадь. Если площадь квадрата уже известна, то найти его стороны, а затем и периметр не составит труда.

Площадь находится по формуле:
S = a²
Это означает, что для того, чтобы вычислить площадь квадрата, нужно умножить длины двух его сторон друг на друга. Как следствие, если знать площадь квадрата, то при извлечении корня из данного значения можно узнать длину стороны квадрата.
Пример: площадь квадрата 36 см², чтобы узнать сторону данного квадрата, необходимо извлечь квадратный корень из значения площади. Таким образом, длина стороны данного квадрата 6 см

Для нахождения периметра квадрата необходимо сложить длины всех его сторон. С помощью формулы это можно выразить так:
P = a+a+a+a.
Если извлечь корень из значения площади квадрата, а затем сложить получившуюся величину 4 раза, то можно найти периметр квадрата.

Пример: Дан квадрат с площадью 49 см². Требуется найти его периметр.
Решение:
Сначала необходимо извлечь корень площади квадрата: √49 = 7 см
Затем, вычислив длину стороны квадрата, можно вычислить и периметр: 7+7+7+7 = 28 см
Ответ: периметр квадрата площадью 49 см² составляет 28 см

1. 

   1

   Recall the formula for the perimeter of a square. For a square of side length S, the perimeter is simply four times the side length: P=4s.^[3]
   

2. 

   2

   Determine the length of one side, and multiply it by 4 to find the perimeter. Depending on the assignment, you may need to measure the side with a ruler, or look at other information on the page to determine the side length. Here are some examples of perimeter calculations:^[4]
   

   • If your square has a side length of 4, then P = 4 * 4, or 16.
   • If your square has a side length of 6, its P = 4 * 6, or 24.

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1. 

   1

   Know the formula for the area of a square. The area of any rectangle (remember, squares are special rectangles) is defined as its base times its height.^[5]
   Since the base and the height of a square are the same length, the area of a square with side length s is s*s, or A = s².

2. 

   2

   Find the square root of the area. The square root of the area will give you the length of one of the sides of the square. For most numbers, you will need to use a calculator to find the square root, by first typing in the value of the area, followed by the square root (√) key.^[6]
   You can also learn to Calculate a Square Root by Hand!

   • If the area of your square is 20, then the side length s =√20, or 4.472.
   • If the area of the square is 25, then s = √25, or 5.

3. 

   3

   Multiply the side length by 4 to find the perimeter. Take the side length s you just calculated and plug it into the perimeter formula, P = 4s. The result will be the perimeter of your square!

   • For the square with area 20 and side length 4.472, the perimeter P = 4 * 4.472, or 17.888.
   • For the square with area 25 and side length 5, P = 4 * 5, or 20.

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1. 

   1

   Understand what an inscribed square is. Inscribed shapes come up fairly often on standardized tests like the GMAT and GRE, so it is important to know what they are. A square inscribed in a circle is a square that is drawn inside of the circle, so that all four vertices (corners) lie on the edge of the circle.^[7]
   

2. 

   2

   Recognize the relationship between the radius of the circle, and the side length of the square. The distance from the center of an inscribed square to each of its corners is equal to the radius of the circle. To find the length of s, we must first imagine cutting the square in half diagonally to form two right triangles. Each of these triangles will have equal sides a and b and hypotenuse c, which we know is equal to two times the radius of the circle, or 2r.^[8]
   

3. 

   3

   Use the Pythagorean Theorem to find the side length of the square. The Pythagorean Theorem states that for any right triangle with sides a and b and hypotenuse c, a² + b² = c². ^[9]
   Since sides a and b are equal (remember, we are still dealing with a square!) and we know that c = 2r, we can write out the equation and simplify the equation to find the side length as follows:

   • a² + a² = (2r)², now simplify the expressions:
   • 2a² = 4r², now divide both sides by 2:
   • a² = 2r², now take the square root of each side:
   • a = √(2r²) = √2r. Our side length s for the inscribed square = √2r.

4. 

   4

   Multiply the side length of the square by four to find the perimeter. In this case, the perimeter of the square P = 4√2r. The perimeter of any square inscribed in a circle with radius r is defined as P = 5.657r!

5. 

   5

   Solve an example equation. Consider a square inscribed in a circle with radius 10. That means that the diagonal of this square = 2(10), or 20. Using the Pythagorean Theorem, we know that 2a² = 20², so 2a² = 400. Now divide both sides in half to find that a² = 200. Then take the square root of each side to find that a = 14.142. Multiply this by 4, and you will find the perimeter of your square: P = 56.57.

   • Notice that you could have found the same thing by simply multiplying the radius, 10, by 5.657. 10 * 5.567 = 56.57, but that might be hard to remember on a test, so it’s better to memorize the process we used to get there.

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Calculator, Practice Problems, and Answers

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• The reason why is because a square has 4 sides.

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