# How to find the area of a square

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Find the area of such a figure as a square, you can even five ways: on the side, perimeter, diagonal, the radii of the inscribed and circumscribed circle.

Instructions

- 1

If the length of the side of the square is known, then its area is equal to the square (second degree) of the side.

Example 1.

Suppose there is a square with a side of 11 mm.

Determine its area.

Decision.

Denote by:

A is the length of the side of the square,

S? Square of the square.

Then:

S = a * a = a? = 11? = 121 mm?

Answer: Square of a square with a side of 11 mm? 121 mm ?.

- 2

If the perimeter of a square is known, then its area is equal to the sixteenth part of the square (second degree) of the perimeter.

It follows from the fact that all (four) sides of the square have the same length.

Example 2.

Suppose there is a square with a perimeter of 12 mm.

Determine its area.

Decision.

Denote by:

P is the perimeter of the square,

S? Square of the square.

Then:

S = (Р / 4)? = Р? / 4? = Р? / 16 = 12? / 16 = 144/16 = 9 mm?

Answer: Square of a square with a perimeter of 12 mm? 9 mm ?.

- 3

If the radius of the circumscribed circle is known, then its area is equal to the quad (quadrally multiplied) square (second degree) of the radius.

It follows from the fact that the radius of the inscribed circle is equal to half the length of the side of the square.

Example 3.

Suppose there is a square with a radius of inscribed circle of 12 mm.

Determine its area.

Decision.

Denote by:

R? Radius of inscribed circle,

S? Square of the square,

A is the length of the side of the square.

Then:

S = a? = (2 * r) = 4 * r? = 4 * 12? = 4 * 144 = 576 mm?

Answer: Square of a square with a radius of inscribed circle of 12 mm? 576 mm ?.

- 4

If the radius of a circle circumscribed about a square is known, then its area is twice the square (of the second power) of the radius (multiplied by 2).

It follows from the fact that the radius of the circumscribed circle is half the diameter of the square.

Example 4.

Suppose there is a square with a radius of the circumscribed circle of 12 mm.

Determine its area.

Decision.

Denote by:

R? Radius of the circumscribed circle,

S? Square of the square,

A is the length of the side of the square,

D? Square diagonal

Then:

S = a? = D? / 2 = (2R?) / 2 = 2R? = 2 * 12? = 2 * 144 = 288 mm?

Answer: The square of the square with a radius of the circumscribed circle of 12 mm? 288 mm ?.

- 5

If the diagonal of a square is known, then its area is equal to half the square (second degree) of the length of the diagonal.

It follows from the theorem of Pythagoras.

Example 5.

Suppose there is a square with a diagonal length of 12 mm.

Determine its area.

Decision.

Denote by:

S? Square of the square,

D? The diagonal of the square,

A is the length of the side of the square.

Then, since by the theorem of Pythagoras: a? + A? = D?

S = a? = D? / 2 = 12? / 2 = 144/2 = 72 mm?

Answer: Square square with a diagonal of 12 mm? 72 mm ?.