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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.

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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.