A regular Pentagon is such that its vertices lie on the circumference of radius 4.5cm . Find the length of a side of the Pentagon to the nearest mm

Answer 1

The radii to two consecutive vertices form an angle of 72°.

s² = (4.5)² + (4.5)² - (4.5)(4.5) cos 72°

Answer 2

radius of 4.5 cm of a circle going through a pentagon's vertices

what is the length of a side to nearest mm?

Assume this is a regular pentagon with equal sides and equal interior angles.

(n-2)180 = (5-2)180 = 3(180)= 540 degrees total of all 5 interior angles

each angle = 540/5 = 108 degrees

construct a triangle with angles 54, 54 (half of the 108) and 72 (180-108), and two sides =4.5, each oppsosite 54 degrees. find the 3rd side

using either with the law of sines or cosines

law of cosines is a generalized Pythagorean Theorem

It helps to draw a picture and lines from the Pentagon center to two adjoining vertices

c^2 = a^2 + b^2 -2abCosC

= 4.5^2 +4.5^2 -2(4.5)(4.5)Cos72

=2(20.25) -2(20.25)Cos72

=40.5 -40.5Cos72

=40.5- 40.5(.309)

=40.5 -12.515

= 27.9848

c =sqr27.9855 = 5.29 cm = 52.9 mm = 53 mm rounded off to nearest integer

or use the law of sines

c/sinC = a/sinA

C=72, a=4.5, A =54

c= asinC/sinA = 4.5sin72/sin54 = 4.5(.951)/.809 = 5.29 cm = 52.9 mm = 53 mm rounded off