Question

The pattern of a soccer ball contains regular hexagons and regular pentagons. The figure below shows what a section of the pattern would look like on a flat surface. What is the measure of each gap between the hexagons in degrees?​

Answer 1

In this pattern, there are 12 regular pentagons and 20 regular hexagons. Each hexagon shares a vertex with three pentagons and each pentagon shares a vertex with five hexagons.

To find the measure of each gap between the hexagons, we can use the fact that the sum of the angles around any vertex in a regular polygon is always 360 degrees. Let x be the measure of the angle between two adjacent pentagons, and y be the measure of each angle at the center of a hexagon.

At each vertex of the pattern, there are three pentagons and three hexagons meeting. Thus, we have:

3(108) + 3y = 360

Simplifying, we get:

324 + 3y = 360

3y = 36

y = 12

Therefore, each angle at the center of a hexagon measures 12 degrees. Since there are six angles around the center of a hexagon, the total angle around the center of a hexagon is 6(12) = 72 degrees.

To find the measure of each gap between the hexagons, we need to subtract the angle of the hexagon from 180 degrees (since the sum of the angles of a triangle is 180 degrees). Thus, the measure of each gap between the hexagons is:

180 - 72 = 108 degrees

Explanation: