Day 43 – regular hexagon inscribed in a circle

Introduction In our previous lesson we learnt how to construct an equilateral triangle inscribed in a circle. We are going to make use of the same idea to construct a regular hexagon inscribed in a circle. A regular hexagon can also be inscribed in a circle in the same way an equilateral triangle was inscribed in a circle.

Vocabulary Hexagon A polygon having six sides Regular hexagon A hexagon having all its sides congruent and all its angles equal. Inscribed polygon A polygon drawn inside another plane figure such that the vertices of the polygon touch the edge of the plane figure, in this case a circle.

Inscribe To draw construct a plane figure inside another figure in such a way that all the vertices of the interior figure touch the exterior figure.

Constructing a regular hexagon inscribed in a circle When a regular hexagon is constructed inside a circle such that all its vertices touch the edge of the circle, we say that the hexagon has been inscribed in the circle. Our tools of construction still remain the compass and straightedge. All the vertices of the regular hexagon are the same distance from the center of the circle because they are all on the circle.

To construct a regular hexagon inscribed in a circle: 1. We draw a circle with center O of a convenient radius using a compass as shown below. Note that we do not change the radius of the compass in the steps to follow. O

2. We use any point on the circle as the center and make an arc on the circle as shown below.

3. We use the intersection point of the arc and the circle as our new center and make another arc on the circle as shown below. Remember that we are using the same radius. O

4. We repeat the process, using the new intersection points constructed as the new centers with the same radius four times until we draw a total of 6 arcs as shown below. O

5. We then use a straightedge to join each of the six intersection points to form the regular hexagon as shown below. We can use a protractor to ascertain that all the six angles are congruent and a ruler to verify that the all the six sides have equal length. O

Example Using a compass and a straightedge only, draw a circle of a convenient radius then construct a regular hexagon inscribed in the circle.

Solution We draw the circle then using the same radius we make six arcs on the circle. We join the intersection points of the arcs using a straightedge to form the regular hexagon as shown below.

homework Use a compass and a straightedge to construct the largest possible regular hexagon that will fit in the circle O shown below. O

Answers to homework The largest possible regular hexagon drawn is the inscribed regular hexagon. We use the center to get the radius.

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