How to find the square area?

Some of us simply skipped math in school, someone was ill, and someone forgot the prescription of school years, but anyway, sooner or later the question arises: "How to find the area of ​​a square?"

The most basic formula for how to find the area of ​​a square:

where:

  • S is the area of ​​a square,
  • and - the side of the square.

Since all sides of the square are equal, the square of the square is the side of the square. For example, we know that the length of the side of a square is 4 cm. Then, by the formula S = a2it turns out: S = 42= 16 (cm2).

Another way to find the area of ​​a square is along the perimeter. The perimeter of the square (P) is equal to the sum of all sides of the square, and since all sides of the square are equal, it has the following formula:

where:

  • P - the perimeter of the square
  • and - the side of the square.

Thus, if we know the perimeter of a square, we can calculate its area using the following formula:

S = (P / 4)2

Dividing the perimeter by 4, we obtain the length of one side of the square, after which it is easy to calculate the area using the first formula.

You can also find the area of ​​the square, if you know the length of its diagonal.The features of a square, such as a geometric figure, are such that its diagonals (a segment drawn between non-adjacent vertices of a square) divide the square into two rectangular and isosceles triangles. A right triangle is such a triangle that has a right angle, and we know that the square has all the angles right. An isosceles triangle is a triangle with two sides equal. The diagonals of a square are at the same time the bisectors of its angles. The bisector is a ray that divides the angle in half.

By the Pythagorean theorem, it is known that the square of the hypotenuse is equal to the sum of the squares of the legs:

from2= b2+ a2

But since our legs are equal, the formula will have the following form:

from2= a2+ a2= 2a2

So:

from2= 2a2

In our case, the hypotenuse is the diagonal of the square (c = d), and the legs are the side (b, e = a). We have:

d2= 2a2

From the above formula, you can derive the formula for finding the leg (side of the square):

a = √d2/2

Substitute this value in the first formula:

S = (√d2/2)2

We reduce the values ​​of the root and the second degree and obtain the formula:

S = d2/2

For example, if the diagonal is 8 cm, then the area of ​​the square is:

S = 82/ 2 = 32 (see).

Another formula for finding the area of ​​a square is along the radius of the inscribed (r) and circumscribed (R) circle.

The inscribed circle is a circle that touches the middle of each side of the square and has a radius equal to half the middle of the side:

r = a / 2

A circumcircle is a circle that touches the top of each corner of a square:

R = d / 2

Thus, to find the area of ​​a square with the help of the radius of the inscribed circle, we obtain the following formula:

S = (2r)2=22* r2= 4r2

S = 4r2

For example, if the radius of the inscribed circle is 3 cm, then

S = 4 * 32= 4 * 9 = 36 (see).

To find the square of the square using the radius of the circumscribed circle, we obtain the following formula:

S = d2/ 2 = 2R2/2=(22* R2) / 2 = 2R2

S = 2R2

Thus, if the radius of the circumcircle is 4, then by the formula:

S = 2 * 42= 2 * 16 = 32 (cm).

Here are all the ways how to find the area of ​​a square, the formulas you also had the opportunity to derive yourself. Successful decisions!



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