If the centre of the pentagon is joined to the vertices, the regular pentagon is divided into 5 isosceles triangles. The apothem of the pentagon is the altitude (height) of each isosceles triangle. The side of the pentagon is the length of the base of each isosceles triangle.

Calculate the area of one of the isosceles triangles and then multiply by 5.

Edit:The data given in the question is consistent, to one decimal place. If you've learned trigonometry you should attempt to show that, in fact, if the sidelength of the regular pentagon is 5.8' then the apothem is actually 3.9915' (correct to four decimal places).