Can Someone help me with these geometry questions about regular polygons?

Question

1.) Find the lengths of the apothem and the radius of a regular
hexagon whose sides have length 6 cm.

2.) Find the lengths of the side and the radius of an
equilateral triangle whose apothem’s length is 8 ft.

3.) Find the lengths of the side and the radius of a regular
hexagon whose apothem’s length is 10 m.

4.) Find the measure of the central angle of a regular polygon of
      a) 8 sides. c) 9 sides.
      b) 10 sides. d) 12 sides.

 

Can someone please help with setup and not just the answer? Thanks!

Answer 1

Area of polygon = 1/2 x Perimeter x Apothem

Apothem =  2 x Area/ Perimeter

 

Use the above formula to find the apothem.

Radius is associated with circles. I am not sure which circle (inscribed or circumscribed?). If it is the circumscribed, then the radius = length of side of hexagon. If it is inscribed, then the radius = apothem.

 

Area of equilateral triangle = (S^2)(√3)/4  where S = length of side

Regular hexagon can be decomposed into 6 equilateral triangles.

 

Apothem = 2 x Area/ Perimeter

Apothem = 2 x (S^2)(√3)/4  / 6S = (S√3)/12  for a hexagon only.

 

Central angle = 360 / n  where n = number of sides of the polygon