O perimeter of the square is the total measurement of the contour of this figure. It represents the sum of the sides of the square, which, as they are all equal, is equivalent to four times the measurement of one of the sides. From the measurement of the diameter or area of the square it is possible to find the measurement of its side and, thus, the measurement of its perimeter.
If a square is inscribed in a circle, it is possible to find the measurement of the side of the square by measuring the radius of the circle.
Read too: How to calculate the area of polygons
Summary about the perimeter of the square
- The perimeter of the square is the sum of the measurements of its four sides.
- One sided square The has a perimeter given by \(P=4a\).
- The diagonal of a side square The It is given by \(d=a\sqrt2\).
- The area of a square The is calculated by \(A=a^2\).
- Side measurement The of a square inscribed in a circle of radius R is found by the relation \(R=\frac{a\sqrt2}{2}\).
How do you calculate the perimeter of a square?
The perimeter of the square is the measurement of the contour of that figure, that is, it is the sum of the measurements of its sidess. Therefore, to calculate the perimeter of the square it is necessary to know the measurement of one of its sides.
Imagine a square with a side measuring The. As its sides have the same measurement, the perimeter of this square is equal to:
\(\mathbf{Perimeter \ of\ square}=a+a+a+a=4\cdot a\)
Example:
What is the perimeter of a square whose side measures 5 cm?
\(Perimeter\ of\ square=5+5+5+5=4\cdot 5=20 cm\)
How to calculate with unknown sides
There are situations in which the side measurement of a square is not informed. In these cases, other information about the square can be used to determine the size of its side and, finally, calculate your perimeter.
The two most common pieces of information related to the side of a square are the area and the diagonal of that figure. A square with side measurement The It has the following area and diagonal measurement:
Example:
What is the perimeter of a square whose diagonal measures \(4\sqrt2\ cm\)?
The diagonal d of a side square The has the following diagonal measurement:
\(Diagonal\ of\ square: d=a\sqrt2\)
Therefore, a square whose diagonal measures \(4\sqrt2\ cm\) It has the following side measurement:
\(a\sqrt2=4\sqrt2\ cm\)
\(a=4\ cm\)
Thus, the perimeter of this square is given by:
\(Perimeter\ of\ square=4\cdot a=4\cdot 4 cm=16 cm\)
Another way to find the measurement of the sides of a square and subsequently its perimeter is by measuring the area of that figure.
Area of the square
The area of the square refers to the region occupied by this figure. To find this measurement, you need to square the measurement of the side of the square.
Thus, a square with a side measuring The has the following area:
\(Area\ of\ square=(side)^2=a^2\)
Example:
What is the perimeter of a square whose area measures 4cm2?
As seen, the area of a square is equal to the square of its side. Thus, if a square has a side measuring The, then:
\(a^2=4\ cm^2\ \)
\(a=\pm\sqrt{4\ cm^2}\)
\(a=\pm2\ cm\)
Since the side length of the square cannot be negative, this square has side length a=2 cm. Therefore, the perimeter of this square is given by:
\(Perimeter\ of\ square=4\cdot a=4\cdot 2 cm=8 cm\)
How do you calculate the perimeter of the square inscribed in a circle?
There may be situations where a square is inscribed in a circle. In this case, with the information about the radius of the circle, it is possible to discover the measurement of the side of the square and, thus, calculate its perimeter.
When a square is inscribed in a circle, the center of the two images is the same. Like this, The radius of the circle will be half the size of the diagonal of the square.
\(R=\frac{d}{2}=\frac{a\sqrt2}{2}\)
Therefore, the radius R of the circumference and the side The of a square inscribed to it fulfill the relationship:
\(R=\frac{a\sqrt2}{2}\)
Example:
What is the perimeter of a square that is inscribed in a circle whose radius measures \(3\sqrt2\ cm\)?
Firstly, through the radius of the circle lies the side of the square:
\(R=\frac{a\sqrt2}{2}\)
\(3\sqrt2=\frac{a\sqrt2}{2}\)
\(2\cdot3\sqrt2=a\sqrt2\)
\(\frac{6\sqrt2}{\sqrt2}=a\)
\(a=6\ cm\)
Thus, the perimeter of this square of side 6cm it's the same as
\(Perimeter\ of\ square=4\cdot a=4\cdot 6 cm=24 cm\)
Read too:Geometric figure congruence criteria
Solved exercises on the perimeter of the square
Question 1
A farmer will fence a square-shaped piece of land. He knows he needs 9 m of wire to fence only one side of the land. How many meters of wire does he need to surround the entire land, this measurement being the perimeter of the land?
a) 9 m
b) 18 m
c) 27 m
d) 36 m
Resolution
Knowing that one side of the land measures the equivalent of 9 m, to surround the perimeter of the entire square plot you will need:
\(Perimeter\ of\ the\ terrain\ square=4\cdot9 m=36 m\)
Therefore, it is necessary 36 m of wire.
The correct alternative is alternative d).
Question 2
A teacher asked her students to draw a square that she had 100 cm2 of area. What should be the perimeter of the square drawn by the students?
a) 10 cm
b) 25 cm
c) 40 cm
d) 100 cm
Resolution
Knowing the area of the square, you can find the length of its side. The through the relationship:
\(a^2=100\ cm^2\ \)
\(a=\pm\sqrt{100\ cm^2}\)
\(a=\pm10\ cm\)
Since the side measurement of the square must be positive, then the side of the square must measure 10cm .
Therefore, the perimeter of this square is equal to
\(Perimeter\ of \ land\ square=4\cdot10 cm=40 cm\)
The correct alternative is option c).
Sources:
REZENDE, E.Q.F.; QUEIROZ, M. L. B. in. Flat Euclidean Geometry: and geometric constructions. 2nd ed. Campinas: Unicamp, 2008.
SAMPAIO, Fausto Arnaud. Mathematics trails, 7th year: elementary school, final years. 1. ed. São Paulo: Saraiva, 2018.
