How many hexagons are there on a football?

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How many hexagons are there on a football?

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Vikas Deshmane · Answer 1 · 2020-05-16T05:25:45+0000

Clarifying question:
No Clarification required - A standard soccer spherical ball. And The ball is spherical in shape and formed only by a combination of the Pentagons and the Hexagons - no other shape is involved.

Given Numbers/ Assumptions:

The Soccer ball has an approximate diameter of 22 cms
Each Pentagon is surrounded by the 5 hexagons along all of its 5 edges of Pentagon
Each hexagon is surrounded by 3 other pentagons

Equation :

The Number of Hegaxagons = nH
The Number of Pentagons = nP
A Pentagon is surrounded by 6 hexagons. There fore then nH = nP * 6
However, each Hexagon is surrounded by the 3 pentagons - Hence thrice counted to we need to divide by 3
nH = nP*6/3
nH = nP *2

The surface area of the Spherical ball = Area of a Hexagon (aH) * The Number of Hexagons + Area of the Pentagon(aP) * Number of Pentagons (nP)

Area Of Heaxagoan (aH ) = six equilateral triangles contained = 3 * sqrt(3) * Sqr(Side) /2 = 2.60 * Sqr(side)
Area of Pentagon = 1.6845 * Sqr (side)
Surface Area of spehere = 4 *PI * sqr (Radius)
Approximately the side of the Hexagoan and the pentagoan could be around 3.5 cms

4 * PI * Sqr(11) = 2.60 * Sqr(3.5) * nP*3+ 1.68 * sqr (3.5) * nP

1519 = 95.55 nP +20.58 nP
nP = 1519 /116.13

nP = 13 Therefor the Number of Pentagons = 13 and the Number of Hexagons = 26 approximately

Sanity Check:

The Actual number of the Pentagons =12 and the Actual number of Hexagons = 20. I believe the framework followed should be good enough to convince the interviewer even if the answer is a bit far from correct.