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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.

## 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?
• 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.
• 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?
• 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.
• 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.
• A regular hexagon has all of its sides being of the same length and all of its angles being of the same measure.
• Is there any way to construct a regular octagon inscribed in a circle?
• 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.
• What was the point of that initial long line through the circle? It never seemed to really get used...
• 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.
• 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."
• 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.
• 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!!