Exercises: Construct regular polygons in circle - Common Core Math - HS Geometry

A

Warm-Up

1.

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

A.

45°

B.

60°

C.

90°

D.

120°

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

2.

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 $OAB$ ?

A.

Scalene

B.

Isosceles

C.

Equilateral

D.

Right

3.

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

A.

All vertices are at the center of the circle

B.

All vertices lie on the circle

C.

All sides of the polygon are diameters of the circle

D.

All angles of the polygon are right angles

B

Fluency Practice

C

Varied Practice

D

Word Problems

$A square inscribed inside a circle of radius 8 meters, with one side labeled as unknown$

1.

A circular garden has a radius of 8 meters. A landscape designer plans to install a square stone path along the perimeter of the largest square that fits exactly inside the garden (inscribed in the circle).

What is the side length of this square, to the nearest tenth of a meter?

2.

A tile artist wants to create a regular hexagonal tile that fits exactly inside a circular ceramic mold of radius 6 cm (meaning all six vertices of the hexagon touch the edge of the mold). She plans to cut the tile so each side is 3 cm, arguing that the sides should be half the radius. Her classmate disagrees.

Who is correct? State the correct side length and explain why using the geometric property of the hexagon construction.

E

Error Analysis

$Hexagon construction diagram showing three correctly placed vertices, then a compass adjustment, then three incorrectly placed vertices$

1.

During a hexagon construction, a student marks the starting point $A$ on the circle and correctly places vertex $B$ with the compass set to radius $r$ . After marking $B$ , she notices the remaining arc looks 'too large' and shortens the compass width before marking vertices $C$ , $D$ , $E$ , and $F$ .

Which statement best identifies the student's error?

A.

The compass width must remain equal to the radius throughout; changing it breaks the equilateral triangle property that makes the construction work

B.

She should have started at a different point $A$ — the arc appears large only because of the starting position

C.

She should have used the diameter setting instead of the radius, which would eliminate the need for adjustments

D.

Physical constructions always accumulate errors, so adjusting the compass to compensate is an acceptable technique

$Circle with two perpendicular chords that do not pass through the center, with their four endpoints connected into a quadrilateral$

2.

A student claims: 'I drew two perpendicular chords inside my circle and connected their four endpoints. Since the chords are perpendicular, I have constructed an inscribed square.' The diagram shows the two perpendicular chords do not pass through the center of the circle.

Which statement correctly identifies the error in the student's claim?

A.

The chords must pass through the center of the circle (they must be diameters) to guarantee four equally spaced vertices

B.

Perpendicular chords always produce a square — the student just needs to label the vertices correctly

C.

The student should measure the arc lengths between the four points to verify they are equal before connecting them

D.

The chords should be drawn at 45° to each other, not 90°, to produce a square

F

Challenge

$Hexagon ABCDEF inscribed in a circle of radius 10, with dashed diagonal AC forming one side of the inscribed equilateral triangle, and an unlabeled arc at center O between radii OA and OC$

1.

A regular hexagon $ABCDEF$ is inscribed in a circle of radius 10. Vertices $A$ , $C$ , and $E$ are connected to form an inscribed equilateral triangle. Using the Law of Cosines (or another method), find the exact length of diagonal $AC$ . Express your answer in simplest radical form.

2.