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paulb203

Member

Registered: 2023-02-24

Posts: 345

pi for a hexagon

Is the ratio of the 'diameter' (corner to furthest corner) of a hexagon to its perimeter 3?

pi for a circle = 3.14

pi for a square (calling that sqi) = 4

pi for a hexagon (calling that hxi) = 3?

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Prioritise. Persevere. No pain, no gain.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: pi for a hexagon

You're talking about a regular hexagon I think; so the answer is yes.  You can divide the hexagon into 6 equilateral triangles each with side = the same as for the hexagon itself.  Let's say side = a. Then perimeter = 6a and diagonal = 2a.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

paulb203

Member

Registered: 2023-02-24

Posts: 345

Re: pi for a hexagon

You're talking about a regular hexagon I think; so the answer is yes.  You can divide the hexagon into 6 equilateral triangles each with side = the same as for the hexagon itself.  Let's say side = a. Then perimeter = 6a and diagonal = 2a.

Bob

Thanks, Bob, but someone elsewhere has pointed out to me an error in my thinking.
I've been thinking about this in relation to my sqi ar ^2 idea.
But with sqi ar ^2 I was using the centre of the square to the centre of a side for the 'radius'. To be consistent I should be using that for the 'radius' of the regular hexagon

I found out that's called the apothem (which means I can dispense with the 'ar' for 'radius' in my formula and now use 'a' for apothem.
So,
sqi(a^2)
And,
hxi(a^2)
respectively, are the formulae

But I got 2 different answers for the value of hxi

Because I got 2 different answers for the area of the regular hexagon (sides 6cm)

I got 2 different formulae for the area online;

(3√ 3/(2))*a^2 where a=length of sides

and

0.5(a)(P) where a=apothem and P= perimeter

apothem seems to equal 5.5cm (?)

---

Prioritise. Persevere. No pain, no gain.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: pi for a hexagon

Starting with the hexagon divided into 6 equilateral triangles.

Let side = a and apothem = h. Note this is the height of a triangle.

Area of a triangle = half base times height = 0.5 ah.

So area of hexagon = 6 times 0.5 ah = 3ah.  But perimenter = P = 6a so area = 0.5 times 6ah = 0.5 hP.

Using Pythag on a half triangle h = √ [a^2 - (0.5a)^2] = √3 a/2

so area = 3ah = 3a .√3.a/2 = 3√3 a^2 /2.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob