Calculate the area of a square: by side, diagonal, perimeter
The square area is a basic concept, thanks to which you can easily calculate the consumption of materials for repairs, calculate the correct dimensions of furniture when measuring a room, understand how much fertilizer and seeds are needed to plant important crops in a huge field.
The given square area formulas are used by builders, furniture manufacturers, and representatives of agriculture.
What is a square?
A square is a regular rectangle with equal sides. Each angle of the figure is 90⁰. A square refers to simple geometric shapes located on a plane. There are several ways to find the area of a square: diagonally, along the side, along the perimeter.
Area formulas, calculation examples
The area of a simple figure is a positive value that has the following properties:
- Equal geometric figures have equal areas.
- If a simple figure is divided into several parts, its total area will always be equal to the sum of the areas of all elements.
- The area of a square is always equal to one if its side corresponds to the unit of measurement.
By side
In geometry, the area is always denoted as S, and small Latin letters (for example, a and b) are the sides of a simple figure.
The basis for calculating the area of any rectangle along the side is a simple formula: S=ab, but in the case of a square, the formula is converted to S = a² because the two sides are the same length.
This implies that the area of a square is equal to the square of its side.
Example 1: Given a square whose side is 5 cm. What is the area?
Decision: S=5²=25cm
Example 2: The side of the figure is 3 cm. Find the area.
Decision: S=3²=9 cm
Diagonally
Another option to find the area is to make calculations relative to the diagonal of the figure (d). True, for this you first need to find the length of the diagonal itself. It is known that the diagonal divides the square into two isosceles triangles. This means that calculations can be carried out according to the well-known Pythagorean theorem, where the sides of the square will act as the legs, and the diagonal itself will be the hypotenuse.
The calculation of the area along the diagonal is made according to the principle: the area of the square is equal to the square of the length of the diagonal (calculated by the Pythagorean theorem) and divided by two.
Example: Given a square whose diagonal is 10 cm. How to calculate the area?
Decision: According to the formula above, the calculations are made as follows: S = 10² / 2 = 100/2 = 50 cm²
around the perimeter
The perimeter is the sum of all the lengths of the sides of a square. The perimeter is denoted by the Latin letter P. Taking into account the definition of a square, we obtain a universal formula for calculating the perimeter for an equilateral quadrangle: P = 4a. That is, the perimeter of a square is four times the length of the side.
Calculating the area of a square relative to the sum of all sides is necessary if only the value of the perimeter is specified in the problem. Knowing the formula for calculating the perimeter, it is very easy to find the area.
If a P = 4a, then a = P/4. Next, you already need to use the formula for calculating the area by side.
Example: Let a square be given with a perimeter of 100 mm. What is the area?
Decision: The side of the square will be 100/4 = 25 mm. Well, the area of the square is further calculated by the formula, where the area of the square is equal to the square of the sides. That is, S = 25² = 625 mm²
Area of a square inscribed in a circle
This option is used as a consequence of the formula obtained earlier (diagonal calculation). According to mathematical data, the diameter of the circle will be exactly equal to the diagonal of the square. Therefore, in order to quickly calculate the area of an equilateral quadrangle, it will be enough to know the diameter of the circle. And then the already known formula is used: S = d²/2
Typical task: for example, given a circle with a diagonal of 8 cm and a square is inscribed in it. What is the area of the quadrilateral?
Correct solution: S = 8²/2 = 64/2 = 32 cm²
