The distance from the centre of a regular pentagon to each of its vertices is 6cm.Calculate the perimeter of the pentagon

Question

I don't really understand this :(

Answer 1

draw a diagram

divide the pentagon into 5 congruent triangles

choose one of the triangles

draw the apothem which will be the height of the triangle

each central angle of each triangle will be 360/5=72°

the apothem divides this 72° angle into two 36° angles

you now have a right triangle with the hypotenuse equal to 6, the apothem, a, is unknown, and the other leg is half of one of the 5 sides

find half of one of the sides using the sine function

sine=opposite/ hypotenuse

sine36°=opposite/6 where opposite=half of one of the sides

opposite=6*sin36°

opposite=6*0.587785

opposite=3.52671

2*3.52671=7.05342

one side of the pentagon is equal to 7.05342 cm (as are all of the other sides)

the perimeter is 5*7.05342=35.2671 cm

Answer 2

You know that the regular pentagon is made up of 5 isosceles triangles that join in the middle. The apex angle of these triangles is 360°/5 = 72°. If you divide one of these triangles in half splitting it down the middle, you make a right triangle with hypotenuse of 6 and an angle of 36° at the top. The vertical is the apothem and the base is 1/2(s) where s is the side:

General formulas derived from this approach for n-gon and a length to vertex of r (also called the radius of the circumscribed circle) is

apothem a = rcos(360/2n) Note: A = 1/2(nas) = 1/2(aP) Area is the sum of the triangle areas

s = 2rsin(360/2n) Note P = ns (What you want)

In your case, n = 5 and r = 6

Take care.