Inscribed Polygon Definition & Regular Hexagon Construction
An inscribed polygon has all its vertices on the edge of a circle, with its interior entirely inside the circle. In Saxon Math, Course 2, Chapter 2, Grade 7 students learn to construct a regular hexagon inscribed in a circle using only a compass and straightedge. The key insight: a circle's radius fits exactly six times around its circumference as a chord. Set the compass to the circle's radius, walk it around the edge six times to mark six equally spaced points, then connect them to form a perfect regular hexagon.
Key Concepts
Property A polygon is inscribed in a circle if all of its vertices lie exactly on the edge of the circle, and all its interior space is inside the circle. To construct a regular hexagon inscribed in a circle of radius $r$, set your compass width to $r$.
Mark a starting point on the circle, and "walk" the compass around the circumference to create six equally spaced intersection points.
Connecting these six points forms a regular hexagon.
Common Questions
What is an inscribed polygon?
An inscribed polygon is a polygon where all vertices lie exactly on the circle's edge and the entire polygon is inside the circle.
How do you construct a regular hexagon inscribed in a circle?
Draw a circle, then without changing the compass width from the radius, walk the compass around the circle's edge to create six equally spaced marks. Connect the marks with a straightedge to form the hexagon.
Why does the radius fit exactly six times around the circle?
Because the interior angle of a regular hexagon is 120° and the central angle is 60°. Six 60° sectors equal 360°, and each chord equal to the radius creates a 60° central angle.
What makes a hexagon 'regular'?
A regular hexagon has all six sides equal in length and all six interior angles equal (each 120°).
Where is inscribed polygon construction taught in Saxon Math Course 2?
This skill is introduced in Chapter 2 of Saxon Math, Course 2, as part of Grade 7 geometry and construction content.
What tools do you need to construct an inscribed hexagon?
You need a compass set to the radius of the circle and a straightedge (ruler without measurements) to connect the six marked points.
How does constructing an inscribed hexagon relate to real life?
Honeycomb cells, bolt heads, and many tile patterns use regular hexagons. Understanding their geometric properties helps in engineering, architecture, and design.