Find the area of a regular pentagon with a perimeter of 240km

Cut your regular pentagon into 5 isosceles triangles each radiating from the centre.

The cut each of those triangles in half making 10 congruent right angled triangles.

the hypotenuse is the radius of the hypotenuse of the triangles and

One of the sides of the triangle is a half of one side of the pentagon

and the other side of the trianngle is the apothem of the pentagon  I am going to call this side $$a$$

The angle radiating from the centre of each right angled triangle is    (360/5)/2 = 36 degrees

the length of the side opposite the centre angle is     0.5(240/5) = 24km

Now use tan to get the length of the apothem.     you will get     $$a\approx 33.03km$$

$$\\Area = 10* $ area of right angled triangle$\\ =10* (0.5*base * height)\\ =10*0.5*24*33.03\\ \approx 3964 km^2$$

The side must be 240/5 = 48km

We have a "formula" to find the area given the side length....I won't derive it for you, but it is given by:

A = 1.7204 s^2   where s is the side length....so we have

A = 1.7204 (48)^2 = 3963.8 km ^2