Area of a hexagon

[mayapuppies]

Re: Area of a hexagon

 

Come on. HERO is the system of "there are more ways to do it than people who want to". The best we can come up with is a measly five answers?

 

Let's try to make this a little more complicated, then. In addition to calculating the area of this hexagon, also try to provide its volume.

And it is a region of a map so one must include the variances in altitude.

 

Thanks everyone for your help! It is greatly appreciated!!

[Mantis]

Re: Area of a hexagon

 

Actually, that circle would surround the hexagon so the area of the hexagon should be less than that of a circle with the same radius. 6,495.

 

Sanity check THAT!

Sorry, your sanity check has bounced .

 

Using half the side-to-side distance as the radius of a circle gets you a circle INSIDE the hex. Using half the point-to-point distance as the radius of a circle gets you a circle OUTSIDE the hex. The area of the hex is between the areas of these two circles.

 

You can do the same thing with rectangles. A rectangle that is the side-to-side (sts) distance high and side-length (los) wide fits wholly inside the hex. A rectangle that is the side-to-side distance high and point-to-point (ptp) distance wide fits wholly outside the hex. And the actual hex area is exactly half-way between these two measurements (the difference between them is eight triangles of equal size, of which four are inside the hex and four outside).

 

The original hexagon had a sts of 100 miles, so:

 

los: sts * tan30deg = 57.74 miles

ptp: sts / cos30deg = 115.47 miles

 

'Inside' rectangle: sts * los = 5774 miles^2

'Outside' rectangle: sts * ptp = 11547 miles^2

Hex area = (inside + outside ) / 2 = 8660.5 miles^2

 

Sanity check using circles:

'Inside' circle: radius = sts/2, [pi]r^2 = 7854 miles^2 (must be less than calculated hex area)

'Outside' circle: radius = ptp/2, [pi]r^2 = 10472 miles^2 (must be greater than calculated hex area)

 

Okay, does my sanity check have sufficient funds?

[cutsleeve]

Re: Area of a hexagon

 

This is why alot of mathmeticians and geniouses go crazy.

[McCoy]

Re: Area of a hexagon

 

Actually' date=' that circle would surround the hexagon so the area of the hexagon should be less than that of a circle with the same radius. 6,495.[/quote']

Look at prestidigitator's diagram in post #16. He's talking the green circle, you're talking the blue circle.

[McCoy]

Re: Area of a hexagon

 

We're Champions players! Are we really this math illiterate?

 

So far we have

A. >5890

B. 6,450

C. 6,495

D. 6495.1905

E. 7500

F. 8,660

G. 8660.25

H. 11,250

I. 7854

 

Nine answers. Fairly sure A is correct, but somewhat vague. Equally sure that I is correct, and a little more specific.

 

Now if I is correct, B, C, D, E, and H must be wrong. So either F or G must be the correct answer.

 

Let an equilateral triangle have sides of length 1. Drop a perpendicular from the apex to the base, cutting the base in half. Use the pythagorean theorem to compute its length.

That's what I overlooked! The base of my 30-60-90 triangle has to be half the hypotenuse! So the ratio of the sides of my 30-60-90 triangle is 1-SQR3-2. So (50/SQR3)*2 = 57.8 miles on a side.

 

I think that's still too large.

 

Also,

Hexagon

It says area = 3/2 * sqrt(3) * side length^2

 

Also, it says inscribed radius = 1/2 * sqrt(3) * side length

 

Now, the "inscribed radius" is the radius of a circle inscribed in the hex ... and if you think about it, for a regular hexagon that circle touches the hex at the midpoints of the hex sides, so twice the inscribed radius = distance across the flats.

OK, let's try simple algebra.

 

50 miles = 1/2 * sqrt(3) * side length [divide both sides by 1/2]

 

100 miles = sqrt(3) * side length [divide both sides by SQRT(3), about 1.73]

 

57.80 = side length

 

Same answer I got above.

 

So (((57.80 * 50) /2) *6) = 8670 sq mi.

 

We have answer number 10!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

We are so pathetic!

[Mantis]

Re: Area of a hexagon

 

We're Champions players! Are we really this math illiterate?

 

So far we have

A. >5890

B. 6,450

C. 6,495

D. 6495.1905

E. 7500

F. 8,660

G. 8660.25

H. 11,250

I. 7854

 

Nine answers. Fairly sure A is correct, but somewhat vague. Equally sure that I is correct, and a little more specific.

Actually, my answer (I) was 8660.5, what you have quoted was my 'sanity check'. Still a different answer, but the differences between F, G and I and your answer of 8670 are all due to rounding in the course of the calculation. I'd say 8660 miles^2 is close enough (within a square mile at most).

 

YM^2MV

[oberon]

Re: Area of a hexagon

 

sqrt(3)/2 gives the ratio of the perpendicular side to the hypotenuse (0.866)

 

If the hexagon is 100 miles from point to point, the side of each triangle is 50 miles and the height is 0.866*50 or 43.3 miles. The area of each triangle is then 50*43.3/2, and the area of the hexagon (6 triangles) 50*43.3*3=6495 sq.miles.

 

If the hexagon is 100 miles between paralell sides, then the height of each triangle is 50 miles, and each side is 50/0.866 or 57.736 miles. The are of the hexagon then becomes 57.736*50*3=8660.4 sq.miles (actually, 8660.25 if you don't round off during any of the calculations)

 

 

oberon

[Rapier]

Re: Area of a hexagon

 

Oh my gawd. Ok, like I just told the boyfriend..."you heard it once, dont expect to hear it twice...you might read it three or four times but I aint repeating myself."

 

I misread. I thought the 100 miles was from point-to-point (I'm still not sure some post editing didn't happen, but thats another issue). You are correct. I was using the right math and equations and methodology. My base assumptions were incorrect.

 

: hangs head in shame.

 

Sigh.

[Robyn]

Re: Area of a hexagon

 

If the hexagon is 100 miles between paralell sides' date=' then the height of each triangle is 50 miles, and each side is 50/0.866 or 57.736 miles. The are of the hexagon then becomes 57.736*50*3=8660.4 sq.miles (actually, 8660.25 if you don't round off during any of the calculations)[/quote']

 

Good. Now, convert that number (in square miles) to the proper number (in cubic miles)

[AlHazred]

Re: Area of a hexagon

 

Actually, I believe in Alabama, the area of that hexagon has been ruled by law to be 9000 miles...

[SuperKlaus]

Re: Area of a hexagon

 

Paging Stochastic to thread...

[McCoy]

Re: Area of a hexagon

 

sqrt(3)/2 gives the ratio of the perpendicular side to the hypotenuse (0.866)

 

If the hexagon is 100 miles from point to point, the side of each triangle is 50 miles and the height is 0.866*50 or 43.3 miles. The area of each triangle is then 50*43.3/2, and the area of the hexagon (6 triangles) 50*43.3*3=6495 sq.miles.

 

If the hexagon is 100 miles between paralell sides, then the height of each triangle is 50 miles, and each side is 50/0.866 or 57.736 miles. The are of the hexagon then becomes 57.736*50*3=8660.4 sq.miles (actually, 8660.25 if you don't round off during any of the calculations)

 

 

oberon

So far we have

A. >5890

B. 6,450

C. 6,495

D. 6495.1905

E. 7500

F. 8,660

G. 8660.25

H. 8660.4

I. 8660.5

J. 8670

K. 11,250

L. 7854

 

Looks like about 8660 is the concensus.

[teh bunneh]

Re: Area of a hexagon

 

Wait wait, let me get this straight. HERO gamers are arguing about math? Say it ain't so!

[McCoy]

Re: Area of a hexagon

 

Wait wait' date=' let me get this straight. [i']HERO gamers[/i] are arguing about math? Say it ain't so!

It ain't so. Less argueing than floundering helplessly.

[Dale A. Ward]

Re: Area of a hexagon

 

So far we have

A. >5890

B. 6,450

C. 6,495

D. 6495.1905

E. 7500

F. 8,660

G. 8660.25

H. 8660.4

I. 8660.5

J. 8670

K. 11,250

L. 7854

 

Looks like about 8660 is the concensus.

 

Mathematics by consensus?

 

Archimedes and Pythagoras are each whirling dervishly in their graves!!

 

[Manic Typist]

Re: Area of a hexagon

 

Where is Von-D Man?

[Mantis]

Re: Area of a hexagon

 

Wait wait' date=' let me get this straight. [i']HERO gamers[/i] are arguing about math? Say it ain't so!

No, HERO gamers are arguing with Math.

 

And Math is winning. It has a helluva DCV...

[teh bunneh]

Re: Area of a hexagon

 

It ain't so. Less argueing than floundering helplessly.

 

No' date=' HERO gamers are arguing [i']with[/i] Math.

 

And Math is winning. It has a helluva DCV...

 

Ah. Well in that case, all is right in the world. Carry on.

 

Mathematics by consensus?

 

Maybe someone should start a poll?

 

Bill.

(So much for the "You need an advanced degree in mathmatics to understand Hero" argument).

[prestidigitator]

Re: Area of a hexagon

 

Well, approximation is going to make answers a little bit different, and unless we keep things symbolic there is going to have to be at least a little approximation since sqrt(3) is an irrational number.

[Markdoc]

Re: Area of a hexagon

 

Huh - I saw the thread title, noticed there had been several replies and passed since I figured the question had been answered. Today I noticed it had spawned three pages so popped in. I, too, stand in awe at the number of different answers we got.

 

The simple formula is: 3 x square root of three divided by2 times the sqaure of the side-length

 

Using this simple formula the area is 6495.2 square miles *if* the sides are 50 miles long. But the original question asked for the area if it was 100 miles *across*. So the sides *can't* be 50 miles long - 'cos then they'd need to be end on end and your hexagon would be um... a square.

 

So.. what you want to know is "how long is each side" - then you can use the formula above. Triangles was a good place to go - but not equilateral triangles (because then we don't know how long the sides are!). What you want to do is draw a 100 mile long line between two points so that they enclose one angle of the hex. Like this:

 

I\

I/

 

OK, it's crude, I admit, but you get the idea. The two "I"s are each 50 miles, so there's your 100 miles. Now we got one side's length. The "\" and "/" are two of your desired sides. Now, since the sum of the interior angles of a triangle is always 180°, and each interior angle of a regular pentagon measures 120° and the other two angles are equal in size, that means they have got to be 30° each.

 

Now we have one side and all the angles, we can work out the length of the two remaining sides. To calculate the length of a triangles' side you use the formula

 

L1 = L2

sin a1 sin a2

 

where L is length and a is angle. If we know one side length (L1) we can then calulate the other (L2) by rewriting the formula as:

 

L2 = L1 x sin a2 or L2 = 100 x 0.5

sin a1 0.86

 

which is 58.1 miles. look at the diagram of the triangle above - that looks about right. Each side should be more than 50 miles, but not by much.

 

So.... back to the formula above. plug in 58.1 and Yes! It's 8770.1 square miles. The exact same answer as .... nobody

 

OK, That was probably more math than desired, but I wanted to show you how I got the answer. Math by consensus, my ***. Next you'll be comparing evolution to religion or some such nonsense or complaining about textbooks in schools

 

Now - just for future reference, I'm assuming you don't always want to do this whole dance, so.... since the square root of three is about 1.73, the formula for a hex can be simplified to 2.6 times the side length-squared and the side length can be simplified to the "radius" of the hex divided by .86, you can sum the whole thing up as:

 

("width" of the hex divided by 1.72) squared times 2.6

 

For this example: (100/1.72)2 x 2.6 = 8788

 

which will give you a rough answer for area of any hex (it'll be too big but not by much and you can do it on any calculator.

 

cheers, Mark

 

You know, I'm amazed I remembered how to do that. I don't think I've done any trig since 1981

[mayapuppies]

Re: Area of a hexagon

 

Huh - I saw the thread title, noticed there had been several replies and passed since I figured the question had been answered. Today I noticed it had spawned three pages so popped in. I, too, stand in awe at the number of different answers we got.

 

The simple formula is: 3 x square root of three divided by2 times the sqaure of the side-length

 

Using this simple formula the area is 6495.2 square miles *if* the sides are 50 miles long. But the original question asked for the area if it was 100 miles *across*. So the sides *can't* be 50 miles long - 'cos then they'd need to be end on end and your hexagon would be um... a square.

 

So.. what you want to know is "how long is each side" - then you can use the formula above. Triangles was a good place to go - but not equilateral triangles (because then we don't know how long the sides are!). What you want to do is draw a 100 mile long line between two points so that they enclose one angle of the hex. Like this:

 

I\

I/

 

OK, it's crude, I admit, but you get the idea. The two "I"s are each 50 miles, so there's your 100 miles. Now we got one side's length. The "\" and "/" are two of your desired sides. Now, since the sum of the interior angles of a triangle is always 180°, and each interior angle of a regular pentagon measures 120° and the other two angles are equal in size, that means they have got to be 30° each.

 

Now we have one side and all the angles, we can work out the length of the two remaining sides. To calculate the length of a triangles' side you use the formula

 

L1 = L2

sin a1 sin a2

 

where L is length and a is angle. If we know one side length (L1) we can then calulate the other (L2) by rewriting the formula as:

 

L2 = L1 x sin a2 or L2 = 100 x 0.5

sin a1 0.86

 

which is 58.1 miles. look at the diagram of the triangle above - that looks about right. Each side should be more than 50 miles, but not by much.

 

So.... back to the formula above. plug in 58.1 and Yes! It's 8770.1 square miles. The exact same answer as .... nobody

 

OK, That was probably more math than desired, but I wanted to show you how I got the answer. Math by consensus, my ***. Next you'll be comparing evolution to religion or some such nonsense or complaining about textbooks in schools

 

Now - just for future reference, I'm assuming you don't always want to do this whole dance, so.... since the square root of three is about 1.73, the formula for a hex can be simplified to 2.6 times the side length-squared and the side length can be simplified to the "radius" of the hex divided by .86, you can sum the whole thing up as:

 

("width" of the hex divided by 1.72) squared times 2.6

 

For this example: (100/1.72)2 x 2.6 = 8788

 

which will give you a rough answer for area of any hex (it'll be too big but not by much and you can do it on any calculator.

 

cheers, Mark

 

You know, I'm amazed I remembered how to do that. I don't think I've done any trig since 1981

owowowowowow

[Markdoc]

Re: Area of a hexagon

 

owowowowowow

 

Yeah - you know, for some perverse reason, I actually enjoyed doing that. probably revenge for all those teachers and lecturers who did it to me.

 

M.

[prestidigitator]

Re: Area of a hexagon

 

Sure. But since we can (and you did) formulate the problem in terms of right triangles, there's really no need for trig. The Pythagorean Theorem is adequate is adequate for this one. If you want to make things more complex you're welcome to, but you might just be shooting yourself (or others) in the foot.

[Markdoc]

Re: Area of a hexagon

 

Sure. But since we can (and you did) formulate the problem in terms of right triangles' date=' there's really no need for trig. The Pythagorean Theorem is adequate is adequate for this one. If you want to make things more complex you're welcome to, but you might just be shooting yourself (or others) in the foot. [/quote']

 

Sure, you could use a right angle triangle - using two sides at 50 miles and dropping those sides into the vertices of the hex - but no-one suggested that and the math is, in any case, the same (you're just changing the lengths and the angles). I figured it would be more intuitive if I simply put two RA triangles back to back and kept the 100 mile measure, but obviously you could do it the way you suggest, instead.

 

Several people suggested using equilateral triangles. That *seems* intuitive, but it only works if you know the length of the side - and in this problem, we didn't. Some people got around that by assuming the 100 miles was point to point - but that's not (at least as far as I can work out) what was asked for, which is why the number they came up with was too low.

 

cheers, Mark

[Dale A. Ward]

Re: Area of a hexagon

 

Duuuuuh...

 

I've got a good little doggie...