The perimeter of the square is found in a variety of ways

Sometimes a person faces the need to find the perimeter of a square. For example, you need to make a fence around a square section, cover with a wallpaper a square room or decorate the walls of a square dance hall with mirrors. To calculate the amount of necessary material, you need to make special calculations. And here, without knowing how to find the perimeter of the square, you will have to purchase the material "by eye". Okay, if it will be inexpensive wallpaper, but the extra mirrors where then to put? And with a shortage of material, then it is difficult to find an additional one of the same quality.

So, how do you know what the perimeter of a square is equal to? We know that at the square all sides are equal. And if the perimeter is the sum of all sides of the polygon, then the perimeter of the square can be written as (q + q + q + q), where q is the value denoting the length of one side of the square. Naturally, it is most convenient to use multiplication here. So, the perimeter of a square is a quadruple value corresponding to the length of its side or 4q, where q is the side.

But if only the area of a square is known , the perimeter of which you need to know - what should you do in this case? And then everything is very simple! From the known figure, which expresses the area of the square, you need to extract the square root. In this way, the side of the square will be found. Now we need to look for the perimeter of the square by the formula derived above.

Another question if you want to find the perimeter of a square along its diagonal. Here we should recall the theorem of Pythagoras. Consider the square WERT with the diagonal WR. WR divided the square into two rectangular isosceles triangles. If the length of the diagonal is known (conditionally take it as z and the side for u), then the side of the square must be sought from the formula: the square z is equal to twice the square u, from which we conclude: u is equal to the square root extracted from the half of the square of the hypotenuse . Then we increase the result 4 times - that's the perimeter of the square!

Find the side of the square by the radius of the circle inscribed in it. After all, the inscribed circle touches all sides of the square, from which the conclusion is drawn - the diameter of the circle is equal to the length of the side of the square. And diameter - this is known to everyone - doubled radius.

If the radius or diameter of a circle described around a square is known, then we see that all 4 vertices of the square are located on a circle. Hence, the diameter of the circumscribed circle is equal to the length of the diagonal of the square. Having adopted this provision as a given, it is next necessary to calculate the perimeter by the formula for finding the perimeter from its diagonal, considered above.

Sometimes a problem is suggested in which it is necessary to find out what is the perimeter of a square that is inscribed in an isosceles rectangular triangle in such a way that one corner of the square coincides with the right angle of the triangle. Known is the cathetue of this geometric figure. We denote the triangle by WER, where the vertex E is common.

The inscribed square will have the designation ETYU. The ET side lies on the WE side, and the EU side on the ER side. The vertex Y lies on the hypotenuse WR. Looking further at the drawing, we can draw conclusions:

1. WTY is an isosceles triangle, since by hypothesis WER is isosceles, therefore, the angle EWR is equal to 45 degrees, and the resulting triangle is rectangular with an angle at the base of 45 degrees, which allows us to assert its isosceles. Hence it follows that WT = TY.
2. TY = ET as the sides of the square.
3. Following the same algorithm, we derive the following: YU = UR, and UR = EU.
4. The sides of the triangle can be represented as the sum of the segments. EW = ET + TW, and ER = EU + UR.
5. Replacing equal segments, we deduce: EW = ET + TY, and ER = EU + UY.
6. If the perimeter of the inscribed square is expressed by the formula (ET + TY) + (EU + UY), then this can be written differently, referring to the newly derived values of the sides of the triangle as EW + ER. That is, the perimeter of a square inscribed in a right-angled triangle with a coincident right angle is equal to the sum of its legs.

This, of course, is not all options for calculating the perimeter of a square, but only the most common ones. But all of them are based on the fact that the perimeter of a quadrilateral is the summed value of all its sides. And from this you can not escape!