Main content

### Course: Geometry (all content) > Unit 16

Lesson 2: Constructing regular polygons inscribed in circles# Geometric constructions: circle-inscribed regular hexagon

Sal constructs a regular hexagon that is inscribed inside a given circle using compass and straightedge. Created by Sal Khan.

## Want to join the conversation?

- I have heard of equilateral, equiangular, and regular. Once you get to pentagons and up they don't have the same meaning.

You can have a pentagon with all sides the same length but unequal angles.

You can have a pentagon with all angles the same but unequal side lengths.

You can have a pentagon that is both equilateral and equiangular which is a regular pentagon.

But this seems counter intuitive to me. How can a polygon be equilateral and not equiangular?(26 votes)- We have equilateral quadrilaterals that are not necessarily equiangular (they're called rhombuses)

We have equiangular quadrilaterals that are not necessarily equilateral (they're called rectangles).

It is only with triangles that a polygon must be equilateral if it is equiangular and must be equiangular if it is equilateral. All other polygons can be equiangular, equilateral, or both.

I suggest you just make a stick model of an equilateral polygon, notice you can physically alter angles and still be able to fit them together into a polygon.(17 votes)

- i understand why the four sides of the hexagon who serve as radii for the additional circles are equal sides. but how do we know that the 2 remaining side (upper and lower) are equal to the rest of the sides?(8 votes)
- If you were connect the center of the circle to both of the top intersections, you would form to equilateral triangles. A quick calculation tells you the angle between those radii is 60 degrees. Since the radii are equal, you have an isosceles triangle with a vertex of 60 degrees, meaning you have an equilateral triangle. Therefore the segment connecting the top intersections (and the one on the bottom) is equal to the radius of your circle.(10 votes)

- What does it mean to be a regular hexagon? I can't find a video explaining what the meaning of "regular" is in terms of polygons.(8 votes)
- A regular hexagon has all of its sides being of the same length and all of its angles being of the same measure.(4 votes)

- Is there any way to construct a regular octagon inscribed in a circle?(5 votes)
- For an octagon, you basically just need to divide the circumference into eight equal pieces. One possible method (though there's a few ways to do it) is:

1. Construct a diameter.

2. Construct the perpendicular diameter (i.e. the perpendicular bisector of the first diameter).

3. Bisect one of the right angles, and draw another diameter - that gives you four arcs subtended by 45°, two on each side of the circle.

4. Now bisect the other right angle, and draw another dimeter - that's the other four arcs.

5. Now just join up all the points where the diameters intersect the circle.(7 votes)

- What was the point of that initial long line through the circle? It never seemed to really get used...(2 votes)
- I think that was simply to make it easier to decide where to put the the two centers of the circles; after all, they have to be directly across from one another, one center at each point where line and circle intersect.(3 votes)

- Is there a video that can help me with a question I have? I have a question about an assignment and don't really know how to do it. It says "Construct line j through D with j ⊥ l."(2 votes)
- if you're given the radius of the circle is there an easy formula to find the area of the hexagon without using trig?

Thank You.(2 votes)- Hexagons are special so you don't need to use trig. Remember that a hexagon can be split into 6 individual equilateral triangles (a triangle with 3 equivalent angles of 60 degrees). So given the radius, you can find the area of each triangle and add them up. Remember, if you split an equilateral triangle into halves, each is a 30,60,90 triangle - meaning: x,2x,x*3^(1/2).

It's easiest if you don't use a formula as they are very tedious, just use logic!

Hope this helped :)

Happy holidays!!(3 votes)

- I am interested in how to construct a regular pentagon in a circle, our geometry teacher taught me how to do it, but I did not know why it work like that(2 votes)
- Because we draw isosceles triangle of 72 angle .The total num we get of triangles is 5 or we get a pentagon(2 votes)

- divide line segament AB = 7 em two parts in the ratio 4:3(2 votes)
- So if we constructed a hexagon on the first circle, then how do we do it on the 2nd and 3rd one?(2 votes)

## Video transcript

Construct a regular hexagon
inscribed inside the circle. So what I'm going
to do, first, I'm going to draw a
diameter of the circle. And actually, I'm going
to go beyond the diameter of the circle. I'm just going to
draw a line that goes through the center of the
circle and just keeps on going. And I'm going to make it flat,
so it goes directly through, so this one right over
here, goes directly through the center. And now, what I'm
going to do is I'm going to construct
a circle that's the exact same dimensions of
this circle that's already been drawn. So let me put this
one right over here. And let me make it
the same dimensions. And now, what I'm going to do
is I'm going to move it over, so that this new
circle intersects the center of the old circle. And these circles
are the same size. So notice, this center
intersects the old circle. And the new circle
itself intersects the center of the old circle. Now, the reason why
this is interesting, we already know that this
distance, the distance between these two centers,
this is equal to a radius. We also know-- we have a
straight edge-- we also know that this distance right
over here is equal to a radius. It's equal to the radius of
our new circle right over here. We also know that this
distance right over here is equal to a radius
of our old circle. And that they both
have the same radius. So this is a radius
between these two points. Between these two
points is a radius. And then, in between these
two points is a radius. So now, I have constructed
an equilateral triangle. And essentially, you have
to just do this six times. And then I'm going to
have a hexagon inscribed inside the circle. Let me do it again. So we'll go from here to here. This is a radius of
my new circle, which is the same as the
radius of the old circle. And I could go
from here to here. That's the radius
of my old circle. So I have another
equilateral triangle. Radius, radius, radius. Another equilateral triangle. I've just got to do this, I
have to do this four more times. So let me go to my original. Let's see, let me make
sure I can-- well, it's actually going to be
hard for me to-- let me just add another circle here,
to do it on the other side. So if I put the
center of it right-- I want to move it a
little bit, right over-- I want to make it the same size. So oh, that's close enough. And let's see, that
looks pretty close. That's the same size. Now, let me move it over here. Now, I want this center to be
on the circle, right like that. And now, I'm ready to draw some
more equilateral triangles. And really, I don't have to
even draw the inside of it. Now, I see my six
vertices for my hexagon, here, here, here,
here, here, here. And I think you're satisfied
now that you could break this up into six equilateral triangles. So let's do that. So this would be the base of one
of those equilateral triangles. And actually, let me move these. Let me move this
one out of the way. I can move this one
right over here. Because I really just care
about the hexagon itself. And I can move this
right over here. But we know that these are
all the lengths of the radius anyway. Actually, I'm not even having
to change the length there. Then, I have to just connect
one more right down here. So let me add another straight
edge, connect those two points. And I would have done it. I would have constructed my
regular hexagon inscribed in the circle.