Question

What radius of a circle is required to inscribe a regular hexagon with an area of 210.44 cm2 and an apothem of 7.794 cm

Answer 1

We know that

[area of a regular hexagon]=6*[area of one equilateral triangle]
210.44=6*[area of one equilateral triangle]
[area of one equilateral triangle]=210.44/6-----> 35.07 cm²

[area of one equilateral triangle]=b*h/2
h=7.794 cm
b=2*area/h------> b=2*35.07/7.794------>b= 9 cm

the length side of a regular hexagon is 9 cm
applying the Pythagorean theorem
r²=h²+(b/2)²------>r²=7.794²+(4.5)²------> r²=81--------> r=9 cm

this last step was not necessary because the radius is equal to the hexagon side------> (remember the equilateral triangles)

the answer is
the radius is 9 cm