Back to Exercise: Construct regular polygons in circle

Exercises: Construct Regular Polygons Inscribed in a Circle

Grade 9·18 problems·~28 min·Common Core Math - HS Geometry·standard·hsg-co-d-13

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Warm-Up

A regular hexagon is inscribed in a circle. What is the central angle subtended by each side of the hexagon?

45°

60°

90°

120°

$Triangle OAB inside a circle, with all three sides labeled r showing it is equilateral$

Triangle $OAB$ is drawn with center $O$ and two adjacent hexagon vertices $A$ and $B$ on a circle of radius $r$ . All three sides equal $r$ . What type of triangle is $OAB$ ?

Scalene

Isosceles

Equilateral

Right

A polygon is inscribed in a circle. What must be true about all vertices of the polygon?

All vertices are at the center of the circle

All vertices lie on the circle

All sides of the polygon are diameters of the circle

All angles of the polygon are right angles

Fluency Practice

To construct a regular hexagon inscribed in a circle of radius $r$ , you first draw the circle and mark a starting point $A$ . What compass setting do you use to mark the first new vertex $B$ on the circle?

Half the radius ( $r/2$ )

The radius ( $r$ )

The diameter ( $2r$ )

A new setting measured by eye to fit the arc

A student constructs a hexagon by walking the compass around the circle. After marking 3 vertices correctly with the compass set to the radius $r$ , she resets the compass to a shorter width before marking the remaining vertices. Which statement best identifies her error?

She should have started at a different point on the circle

She should not change the compass width; the side of an inscribed regular hexagon always equals the radius

Six arcs will not fit evenly because small errors accumulate, so adjusting is fine

She marked too many vertices — a hexagon only needs 5 arcs

$Circle with center O and horizontal diameter AB, ready for the next construction step$

To construct a square inscribed in a circle, you first draw a diameter $AB$ . What is the correct next step?

Mark two more points using the same compass radius setting as the hexagon construction

Construct the perpendicular bisector of $AB$ , which passes through center $O$

Draw another diameter at exactly 45° to $AB$ by estimating with a protractor

Bisect arc $AB$ by trial and error until four equal arcs are found

A regular hexagon $ABCDEF$ is inscribed in a circle. Which set of vertices, when connected, forms an equilateral triangle inscribed in the same circle?

$A$ , $B$ , $C$ (three consecutive vertices)

$A$ , $C$ , $E$ (every other vertex)

$A$ , $B$ , $D$ (two consecutive, then skipping one)

$A$ , $D$ (two opposite vertices) and the center $O$

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