Sometimes a person faces a close need to find the perimeter of a square. For example, you need to make a fence around a square plot, to wallpaper a square room or to decorate the walls of a square dance hall with mirrors. To calculate the amount of material needed, you need to make special calculations. And here, not knowing how to find the perimeter of a square, you have to acquire material “by eye”. Okay, if it will be inexpensive wallpapers, but where to put extra mirrors? And with a lack of material, it is then quite difficult to pick up an additional one of the same quality.
So, how do you know what the perimeter of a square is? We know that a square has all sides 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 a value that indicates 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 the square is known, the perimeter of which you need to know - what to do in this case? And here it is very simple! From the known figure, which expresses the area of the square, you need to extract the square root. Thus, the value of the side of the square will be found. Now you need to look for the perimeter of the square according to the formula derived above.
Another question is if you need to find the perimeter of a square along its diagonal. Here we should recall the Pythagorean theorem. Consider the WERT square with the diagonal WR. WR divided the square into two right-angled isosceles triangles. If the length of the diagonal is known (we will arbitrarily take it for z, and the side for u), then the value of the side of the square should be sought based on the formula: the square z is equal to twice the square u, whence we conclude: u is the square root extracted from half the square of the hypotenuse . Further, we already increase the result by 4 times - here you have the perimeter of the square!
You can 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, whence the conclusion is drawn - the diameter of the circle is equal to the length of the side of the square. And the diameter - it is known to all - doubled radius.
If the radius or diameter of the circle circumscribed around the square is known, then here we see that all 4 vertices of the square are located on the circle. Therefore, the diameter of the circumscribed circle is equal to the length of the diagonal of the square. Taking this position as a given, then the perimeter should be calculated according to the formula for finding the perimeter along its diagonal, discussed above.
Sometimes a task is proposed in which you need to find out what is the perimeter of a square that is inscribed in an isosceles right triangle so that one corner of the square coincides with the right angle of the triangle. Known is the leg of this geometric figure. We denote the triangle by WER, where the vertex E is common.
The inscribed square will be marked ETYU. The ET side lies on the WE side, and the EU side lies on the ER side. The vertex Y lies on the hypotenuse WR. Considering the drawing further, we can conclude:
- WTY is an isosceles triangle, since by the WER condition it is an isosceles triangle, which means that the angle EWR is 45 degrees, and the resulting triangle is rectangular with an angle at the base of 45 degrees, which allows us to state its isosceles. It follows that WT = TY.
- TY = ET as sides of the square.
- Following the same algorithm, we derive the following: YU = UR, and UR = EU.
- The sides of the triangle can be represented as the sum of the segments. EW = ET + TW, and ER = EU + UR.
- Replacing equal segments, we derive: EW = ET + TY, and ER = EU + UY.
- If the perimeter of an inscribed square is expressed by the formula (ET + TY) + (EU + UY), then this can be written differently, referring to the just derived values of the sides of the triangle, as EW + ER. That is, the perimeter of a square inscribed in a right triangle with a matching right angle will be equal to the sum of its legs.
These, of course, are 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 quadrangle is the summed value of all its sides. And there’s no getting around it!