For a regular hexagon, there are several formulas for computing the area.
For instance, there are area formulas if you have a side length, the inradius (also called the apothem), the circumradius, or the perimeter.
Since we know the area and would like to know the side length, it would seem simplest to use the side-length area formula, and back-solve it for the side length.
Area = (3/2)(√3)s2
where s is the side length.
Substituting in the given area, 54√3 square meters,
54√3 = (3/2)(√3) s2
Divide out √3 from both sides, and flip the sides:
(3/2)s2 = 54
If a=b, then b=a, so flipping sides is a legitimate action. This is called the symmetric property of equality.
Now, multiply both sides of the above equation by (2/3), the reciprocal (or muliplicatve inverse) of (3/2). The result is
s2 = 36
and if we take the square root of both sides, we arrive at
s = 6.
(Since we don't use negative units for measuring distances, we discard the negative square root.)
We need to check this value to make sure we found the correct value..
Area = (3/2)(√3)s^2
= (3/2)(√3)(6)^2
= (3/2)(√3)(36)
= 54√3,
so we have confirmed the numeric part of our answer.
The units will be the square root of meters squared, so our final answer is
6 meters.
Dont ever forget the units!
To find the apothem a, use the following formula:
a = (1/2) × √3 × s = (1/2) × √3 × 6 = 3√3 meters.
The perimeter P is just 6s for a regular hexagon with side length s, so
P = 6s = 6 × (6) = 36 meters.
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If you need a little help remembering whether the apothem is the same thing as the circumradius or the same thing as the inradius, just remember this:
THE APOTHEM IS NOT THE SAME THING AS THE CIRCUMRADIUS.
Note that there is an "m" in apothem, circumcenter, and same. And that's the one it's not the same thing as!
By the way, the inradius is the radius of the largest circle that fits entirely inside a polygon. The circumradius is the radius of the smallest circle that fits around a polygon.
