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Geometry: High School15 chapters | 160 lessons

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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.

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!

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.

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

- Pick a location on the circle's edge for one point of your hexagon, and make a small pencil mark there.
- 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.
- Work your way around the circle, making marks that are equal distances apart.
- Using a straight edge, draw a straight line between each mark and the two marks on either side of it.

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.

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.

Okay, now we'll change gears a little bit and construct a square. A square is a four-sided figure with equal, straight sides and 90-degree internal angles. To construct one, we'll have to use a little different technique than we did with the hexagon. First of all, draw a circle to put your square in. You can use your last one, if you want, but the other marks might be a little confusing, and we don't really need them to construct a square. Once you have your circle, draw a diameter (line that goes through the center and stops at the outside edge on both sides of the circle).

The diameter marks two of the four corners of your square. Now we need the other two corners. To get those, we'll construct a perpendicular line (crosses at 90 degrees), intersecting your diameter at the center of the circle.

- Push the metal point of your compass into the paper at one of the ends of your diameter.
- Set the length slightly larger than the distance from that point to the center.
- Draw an arc near the center of the diameter but just outside the circle.
- Leave your compass setting the same, and move the point to the other end of the diameter.
- Draw an arc that intersects your previous arc. If you can't quite reach, you might have to lengthen the first arc a little bit.
- Connect the dots. Draw a line that connects the circle's center to the place where your two arcs cross. That's a perpendicular line. Make sure your line is long enough to mark the entire diameter (touches the circle's curved edge at both ends).

Almost done! Now, all you have to do is connect the four marks you've constructed on the circle's edge. Notice that where you put your diameters will determine the position of your box.

If you have positioned them like an X, you'd have a square that appears to be sitting flat on a surface. The one we will draw here will look more like a diamond, but you can put your square in any position you want, just by putting your first diameter at a different angle.

Okay, let's draw in our square.

**Constructing** geometric shapes means to draw them using only a compass and a straight-edge. **Inscribing** them inside a circle means to construct them with each outer corner touching the curved edge of the circle. You can construct hexagons and triangles by using the radius (distance from the center to the edge of the circle) as a guide. You can construct a square by using perpendicular diameters. Once you're used to making these, you can draw accurate regular hexagons, equilateral triangles, and squares, anywhere you need them.

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11 in chapter 3 of the course:

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Geometry: High School15 chapters | 160 lessons

- Line Segments & Rays: Definition & Measurement 3:59
- Types of Angles: Vertical, Corresponding, Alternate Interior & Others 10:28
- Geometric Constructions Using Lines and Angles 4:32
- Line Segment Bisection & Midpoint Theorem: Geometric Construction 4:39
- Dividing Line Segments into Equal Parts: Geometric Construction 5:22
- Parallel, Perpendicular and Transverse Lines 6:06
- Constructing Perpendicular Lines in Geometry 3:39
- Constructing an Angle Bisector in Geometry 3:36
- Methods & Tools for Making Geometric Constructions 3:17
- Practice Making Geometric Constructions with Tools 4:17
- Constructing Equilateral Triangles, Squares, and Regular Hexagons Inscribed in Circles 5:00
- Go to High School Geometry: Introduction to Geometric Figures

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