Question

A regular hexagon with side length 4 has the same area as a square. What is the length of the square?

Answer 1

Answer: The length of the square is $6.44$

Step-by-step explanation:

By definition, the area of a Regular polygon can be calculated with the following formula:

$A=\frac{s^2n}{4tan(\frac{180}{n})}$

Where "s" the length of any side of the polygon and "n" is the number of sides .

According to the information given in the exercise, you know that:

$s=4$

Since an hexagon has six sides, you know that:

$n=6$

Therefore, its area is:

$A_h=\frac{(4^2)(6)}{4tan(\frac{180}{6})}\\\\A_h=24\sqrt{3}$

The formula to find the area of a square is:

$A=s^2$

Where "s" is the length of any side of the square.

Since that regular hexagon has the same area as this square, you can substituting the area calculated above into the formula for calculate the area of a square, and then solve for "s".

Then you get:

$24\sqrt{3}=s^2\\\\\sqrt{24\sqrt{3}}=s\\\\s=6.44$

Answer 2

Area of given square is 41.56 so side of square is 6.4 unit (Approx.)

Area of hexagon and square:

What information do we have?

Area of regular hexagon = Area of square

Side of regular hexagon = 4 units

Area of regular hexagon = Area of square

So,

$\frac{3\sqrt{3} }{2}a^{2} = Side^2$

$\frac{3\sqrt{3} }{2}4^{2} = side^2$

24√3 = Side²

Side² = 41.56

Side = 6.4 unit (Approx.)

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