Find the area of ​​a figure such as a square, it is possible even in five ways: on the side of the perimeter, diagonal, radius of the inscribed and circumscribed circle.

instructions

1

If you know the length of the side of the square, its area is equal to the square (second degree) side.
Example 1.
Suppose there is a square of side 11 mm.

Determine its area.
Decision.
Let:

and - the length of the side of the square,

S? area of ​​the square.
Then:
S = a * a = a = 11? = 121 mm?
Answer: The area of ​​a square with sides of 11 mm? 121 mm ?.

2

If you know the perimeter of a square, the surface area is one sixteenth of the square (second power) of the perimeter.
This follows from the fact that all (four) sides of the square are the same length.
Example 2.
Suppose there is a square with a perimeter of 12 mm.

Determine its area.
Decision.
Let:

P - perimeter of the square,

S? area of ​​the square.
Then:
S = (P / 4)? P =? / 4? F =? / 16 = 12? / 16 = 144/16 = 9 mm?
Answer: The area of ​​a square with a perimeter of 12 mm? 9 mm ?.

3

If you know the radius of inscribed circle into a square, the surface area is equal to four times (multiplied by 4) to the square (second power) radius.
This follows from the fact that the radius of the inscribed circle is equal to half the length of the square.
Example 3.
Suppose there is a square with a radius of the inscribed circle of 12 mm.

Determine its area.
Decision.
Let:

r? the radius of the inscribed circle,

S? area of ​​the square,

and - the length of the side of the square.
Then:
S = a? = (2 * r) = 4 * r? = 4 * 12 = 144 * 4 = 576 mm?
Answer: The area of ​​a square with the radius of the inscribed circle of 12 mm? 576 mm ?.

4

If you know the radius of the circle around the square, its area is equal to twice (multiplied by 2) square (second degree) radius.
This follows from the fact that the radius of the circumscribed circle diameter is equal to half of a square.
Example 4.
Suppose there is a square with a radius of 12 mm in circumference.

Determine its area.
Decision.
Let:

R? the radius of the circle,

S? area of ​​the square,

and - the length of the side of the square,

d? square diagonal

Then:
S = a? = D? / 2 = (2R?) / 2 = 2R? = 2 * 12 = 144 * 2 = 288 mm?
Answer: The area of ​​a square with a radius of 12 mm in circumference? 288 mm ?.

5

If you know the diagonal of a square, its area is equal to half the square (second degree), the length of the diagonal.
This follows from the Pythagorean theorem.
Example 5.
Suppose there is a square with a diagonal of 12 mm.

Determine its area.
Decision.
Let:

S? area of ​​the square,

d? diagonal of a square,

and - the length of the side of the square.
Then, as the Pythagorean theorem: a + a = d???
S = and? = D? / 2 = 12? / 2 = 144/2 = 72 mm?
Answer: The area of ​​a square with a diagonal of 12 mm? 72 mm ?.