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### Apothem of a Hexagon

```
Date: 6/11/96 at 0:11:46
From: RS
Subject: Apothem of a hexagon

I need help in finding the apothem of a regular hexagon. Could you
help me by telling me the formula.

```

```
Date: 6/13/96 at 23:48:28
From: Doctor Brian
Subject: Re: Apothem of a hexagon

Well, let's divide the regular hexagon into six isosceles triangles,
and then we'll deal with one of those--they're all congruent anyways,
so the apothem will be the same.

Now, since this is a hexagon, it's not too difficult to show that
these are not just isosceles triangles, but equilateral.  (After all,
the angle at the center of the circle, the vertex angle of the
triangle, is 360/6 = 60)

Lets call the side length s.  Because these are equilateral triangles,
the lines from the center to the vertices are also of length s.

Now draw the apothem.  It will divide the side into two parts
each with length s/2.

From here we use either the Pythagorean theorem (hypotenuse = s and
one leg = s/2) or we use the 30-60-90 triangle relationship to get
that the apothem is sqrt(3) * s / 2

If you've done anything with sine/cosine/tangent, then you may know
that the sqrt(3) part comes from the tangent of 60.  The 60 in
question is half the interior angle of 120.  It's *not* the 60 from
the central angle.  What I'm getting at is that, if you know the
interior angle of the n-gon, you can replace the sqrt(3) with the
corresponding tangent value.

example:  with an octagon, each angle is 135 degrees, so half of that
is 67.5, and the apothem is tan 67.5 * s / 2

anyways, that's a little extra that you might not need.

see ya

--Doctor Brian
```

```
Date: 6/15/96 at 7:30:29
From: Anonymous
Subject: Re: Apothem of a hexagon

Thank you for your reply. Our situation is quite remote as we are a
missionary family working in the ROC with very little access to
English material to assist in our two teenagers' education. They are
currently home-schooled through an American correspondence course
offered through A Beka. So we wish to thank you for you help.
Best Regards Bob
```

```
Date: 6/15/96 at 9:49:48
From: Doctor Ethan
Subject: Re: Geometry

Well please keep sending your questions to us we are thrilled to be

-Doctor Ethan,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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