What is the area of a regular hexagon with a side length of 8.1 yards and an apothem length of 7 yards?

Area of regular hexagon is ##170.1## square yards.

The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. One can also say that it is the line drawn from the center of the polygon that is perpendicular to one of its sides. Its length is the radius of the circle that can be inscribed in the regular polygon.

Here we have been given that side length of regular hexagon is ##8.1## yards, while apothem is ##7## yards. In fact only one of the two is required. As can be seen from following figure, apothem is related to side of the hexagon. In fact, if a two adjacent vertices of a hexagon are joined to the center they form an equilateral triangle and apothem, half the side and the line joining center to the corner of hexagon forms a right triangle with angles ##30^o-60^o-90^o##.

As such, if one side of regular hexagon is ##2s## from we have

##s^2+7^2=(2s)^2## or ##3s^2=49## or ##s=7/sqrt3=4.0415##

and hence side of hexagon is ##2xx4.0415=8.083##.

Let us round it to ##8.1## as indicated in question itself.

Then area of the equilateral triangle formed is ##1/2xx7xx8.1=28.35## square yards

and area of regular hexagon is ##28.35xx6=170.1## square yards.