Question

list shapes and angles used in football ​

Answer 1

Answer:

Given a standard football, regular hexagons and pentagons are drawn on it as shown in the picture. Find out the number of hexagons and pentagons.

Number of Pentagons and Hexagons on a Football 1

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

We can apply Euler Characteristics to find out several Hexagons and Pentagons on a standard Football.

According to Euler Characteristics: For every surface S there exists an integer

 $$\\chi(S)$$

such that whenever a graph G with V vertices and E edges is embedded in S so that there are F faces(regions divided by the graph), we have:

  $$ V - E + F = \\chi(S)$$

For a sphere(shape of a football)

 $$\\chi(S)$$

= 2.

Hence, the equation becomes V – E + F = 2.

Now, let some pentagons P and the number of hexagons H.

Some vertices will be:

6 vertices for each of the hexagons, i.e. 6*H.

5 vertices for each of the pentagons, i.e. 5*P.

But we have counted each vertex thrice, once for each adjacent polygon, following the picture

Number of Pentagons and Hexagons on a Football 2

Hence, number of vertices, V = (6*H + 5*P)/3.

Some edges will be:

6 edges for each of the hexagons, i.e. 6*H.

5 edges for each of the pentagons, i.e. 5*P.

But we have counted each edge twice, once for each adjacent polygon, follow the picture

Number of Pentagons and Hexagons on a Football 3

Hence, number of edges, E = (6*H + 5*P)/2.

Several faces will be:

There are H hexagons and P pentagons, each forming a face. Hence, a total number of faces, F = (H + P).

So, we can write:

  $$ V - E + F = 2 $$ \\\\ $$ \\frac{6*H + 5*P}{3} - \\frac{6*H + 5*P}{2} + (H + P)  = 2$$

after solving this equation we will get P = 12. So, there are 12 Pentagons.

Now the number of Hexagons:

We can see that each pentagon is surrounded by 5 Hexagons. So there should be 5*P hexagons, but we have counted each hexagon thrice for each of its 3 adjacent pentagons. Hence, number of hexagons = 5*P/3 = 5*12/3 = 20.

Hence, there are 20 Hexagons and 12 Pentagons in a standard football.

Step-by-step explanation: