Constructing Equilateral Triangles, Squares, and Regular Hexagons Inscribed in Circles

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  • 0:03 Constructing Figures…
  • 0:51 A Hexagon in a Circle
  • 1:53 An Equilateral…
  • 2:25 A Square Inside a Circle
  • 4:25 Lesson Summary
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Lesson Transcript
Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Constructing an accurate triangle, square, or hexagon within a circle is a matter of a few simple steps, and can be extremely practical. In this lesson, we'll practice the construction of those three shapes.

Constructing Figures in Circles

Triangles, squares, hexagons, and other shapes show up all the time. If you look around your room, you see these shapes dominating most of the objects. In mathematics, we try to understand and create those shapes. We can create accurate shapes by constructing them around the outside or inside of another object, like a circle.

Constructing geometric shapes means to make accurate shapes using a compass and a straight-edge. A fun one to start with is the regular (all sides the same length) hexagon, because the length of each side of your hexagon is the same length as the radius (distance from the center to the curved outer edge) of your circle. This is great, because if you set your compass to that distance, you can just work your way around the outside edge. So, let's make one!

A Hexagon in a Circle

Draw a circle. The size of your circle will define the outside edges of your hexagon, so pick a size you like. Grab your compass, stab your paper right where you want the center of your hexagon, set the radius length between the legs of the compass, and draw.

Circle with center, radius, and curved edge labeled
diagram of a circle

Once you have your circle, we can start the fun stuff:

  1. Pick a location on the circle's edge for one point of your hexagon, and make a small pencil mark there.
  2. Plunge the metal point of your compass into the paper at that mark. Make sure your compass is still set to the radius length, and make another mark on the circle's edge.
  3. Work your way around the circle, making marks that are equal distances apart.
  4. Using a straight edge, draw a straight line between each mark and the two marks on either side of it.

Constructed hexagon
constructed hexagon

There it is! The hexagon is inscribed, which means it's inside the circle and all of its outer corners touch the circle's edge. Okay, now let's move on to inscribing a triangle.

An Equilateral Triangle in a Circle

Now we'll construct an equilateral triangle, which is a closed, three-sided figure with straight sides. All of its sides are the same length, and all of its internal angles are 60 degrees.

We've already done a lot of the work. If you take your circle, with the six marks you made for the hexagon, you can use those same marks to make your triangle. Pick one of the marks to be a corner, then connect that mark to every second mark, around the circle. When you're done, three marks will have lines connecting them, forming a triangle, and the other three will not be part of the triangle.

Constructing a triangle
constructing a triangle

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