Question

A ring of alternating regular pentagon and a square is constructed byIn this pattern. How many pentagons wil there be in the completed ring​

Answer 1

Answer:

No. of pentagons = 10

Step-by-step explanation:

The figure is attached

We know that if there are n sides of a regular polygon then the sum of ll internal angles is given by

$\boxed{S=(2n-4)\times90^\circ}$

And the measure of one internal angle is given by $\boxed{S/n}$

The length of the sides of square and pentagon are equal

Internal angle of one square = 90°

Sum of all internal angles of pentagon = [(2 × 5) - 4]×90°

                                                    = 6 × 90° = 540°

One internal angle of the pentagon = 540°/5 = 108°

Thus, in the ring, one internal angle will be = 360° - (108° + 90°)

                                                                        = 162°

Let the sides of the ring are N

then sum of all the internal angles = (2N - 4) × 90°

(2N - 4) × 90° = 162°N

180°N - 360° = 162°N

⇒ 18°N = 360°

⇒ N = 360°/18° = 20

Thus there are 20 sides in the ring

No. of pentagons = 20/2 = 10

Hope the answer is helpful.