1. Hexagon- Finding missing information

If a hexagon has a radius (center to point of angle) of 6, what is the side of the hexagon?

If a hexagon has a radius (center to point of angle) of 6, what is the area of the hexagon?

If a hexagon is resting on a flat side, and has a total height of 18, what is the length of each side of the hexagon?

If a hexagon is resting on a flat side, and has a total height of 18, what is the area of the hexagon?

How do I calculate these? Thanks for the potential help.

2. I am going to assume you mean a regular hexagon.

Since it is regular, the hexagon can be broken down into 6 equal triangles. It is like cutting up a hexagonal cake, so each slice would have an angle of 360/6 degrees, so 60 degrees. You are told the radius is 6, so you now have a triangle where you know 2 sides (both 6) and the angle between them (60 deg). Hmm what sort of triangle would this be I wonder? Can you guess the answer yet? Now use the cosine rule to work out the opposite edge to the angle, which gives the length of one side of the hexagon. Now multiply it by 6 and you have your circumference.

Hint: the answer is 36 for the circumference of the hexagon, now you can do the working.

Edit: (just re-read your post and you might only be after the length of one side - in which case you dont multiply by 6 and the answer is 6)

For the second part, the same applies, split it into 6 triangles as before. Now find the area of the aforementioned triangle (cut the triangle in half and use pythag to find the height, which should be sqrt(27), then do 0.5*base*height). Now multiply it by 12 (as you cut it in half) and you have the area.

hint: the answer is 18*sqrt(27), now do the working.

So the main idea is to cut the hexagon into 6 equal triangles then calculate on that. I'll leave the other two to you for know, let me know if you need any more help on them.

Si

3. Yes it is a regular Hexagon. "Now use the cosine rule to work out the opposite edge to the angle, which gives the length of one side of the hexagon. Now multiply it by 6 and you have your circumference." Could you explain this in more detail, please? So I need to use (Cos 60)6 ? Also, I am not asked anything about circumferences. How exactly did you get sqrt(27)?

Originally Posted by bananaxxx
I am going to assume you mean a regular hexagon.

Since it is regular, the hexagon can be broken down into 6 equal triangles. It is like cutting up a hexagonal cake, so each slice would have an angle of 360/6 degrees, so 60 degrees. You are told the radius is 6, so you now have a triangle where you know 2 sides (both 6) and the angle between them (60 deg). Hmm what sort of triangle would this be I wonder? Can you guess the answer yet? Now use the cosine rule to work out the opposite edge to the angle, which gives the length of one side of the hexagon. Now multiply it by 6 and you have your circumference.

Hint: the answer is 36 for the circumference of the hexagon, now you can do the working.

Edit: (just re-read your post and you might only be after the length of one side - in which case you dont multiply by 6 and the answer is 6)

For the second part, the same applies, split it into 6 triangles as before. Now find the area of the aforementioned triangle (cut the triangle in half and use pythag to find the height, which should be sqrt(27), then do 0.5*base*height). Now multiply it by 12 (as you cut it in half) and you have the area.

hint: the answer is 18*sqrt(27), now do the working.

So the main idea is to cut the hexagon into 6 equal triangles then calculate on that. I'll leave the other two to you for know, let me know if you need any more help on them.

Si

4. use pythag to find the height.... I also need help with this part. 6^2 + 6^2 = C^2? Also need help with last two. Thank you.

5. The third problem is different in that we are given the height, 18.
In the diagram that is the length of the green line segment.
That is twice the length of the altitude one of the six equilateral triangles.
So what is the length of the side of an equilateral triangle with altitude 9?
Once you have that find the area of the equilateral triangle and multiply by six to get the area of the hexagon.

6. Originally Posted by KevinVM20
Yes it is a regular Hexagon. "Now use the cosine rule to work out the opposite edge to the angle, which gives the length of one side of the hexagon. Now multiply it by 6 and you have your circumference." Could you explain this in more detail, please? So I need to use (Cos 60)6 ? ?
The cosine rule states that $\displaystyle c^2=a^2+b^2-2ab \cos (D)$ as shown in the picture. So you know the values of a,b and D, so you can work out the value of c which is the length of one of the sides of the hexagon .

How exactly did you get sqrt(27)
I divided the triangle into two pieces, so that the vertical makes a right angle with the base line (see picture). This vertical line cuts the base line exactly in half. Now using pythagoras (and the fact that you know the length of the base line from the previous question) i found that $\displaystyle height^2 + 3^2 = 6^2$ so $\displaystyle height = \sqrt{27}$. No you can find the area of the triangle by multiplying $\displaystyle \sqrt{27}$ by the length of the base (6) and then by $\displaystyle \frac{1}{2}$. To now find the area of the whole hexagon you multiply by 6 as there are 6 triangles which make up the hexagon.

7. First question with height of 18

AB|/|AC| = sin(60) = sqrt(3)/2
|AC| = 2 |AB|/sqrt(3) = 18/sqrt(3) = 10.39230484541326376116467804904

I did this math in order to find the side of the Hexagon.

Here is the work I did for the second question with height of 18 :

(9^2)(sqrt of 3)/4=(35.074028853269765193930788415494)(6)

Although I strongly feel that this is correct, my teacher begs to differ. I must somehow include 10.39 in my calculations when finding the area.

Originally Posted by Plato
The third problem is different in that we are given the height, 18.
In the diagram that is the length of the green line segment.
That is twice the length of the altitude one of the six equilateral triangles.
So what is the length of the side of an equilateral triangle with altitude 9?
Once you have that find the area of the equilateral triangle and multiply by six to get the area of the hexagon.

8. Ok total height of 18 means two triangles on top of each other (as plato shows in his/her diagram) both with height 9.

So cut the triangle in half (as I did in my diagram) to form a right angled triangle. The height is 9 and one angle is 30deg as found earlier (half of 60 as the angle is in half too).

So we use tan to find the length of the base.

$\displaystyle \tan(30)=\frac{base}{9}$
so
$\displaystyle 9\tan(30)=base$
so
$\displaystyle base = 9*\frac{1}{\sqrt{(3)}}$

This is the length of HALF the base (as we cut it in half). So we need to double it to get the full length of one of the sides.

So the answer for the third part is
$\displaystyle base = 2*\frac{9}{\sqrt{(3)}} \approx 10.39$

Ok now hou have the length of each side.

To find the area break it into 6 triangles. You know the height of each is 9 and the base is $\displaystyle \approx 10.39$. So do half base times height times 6.

$\displaystyle \frac{10.39}{2}*9*6=280.53$ Which is the total area of the hexagon.

Si