Figure A shows a regular hexagon inscribed in a circle of...

Figure A shows a regular hexagon inscribed in a circle of radius $1 .$ Figure $\mathrm{B}$ shows a regular heptagon (seven-sided polygon) inscribed in a circle of radius $1 .$ In Figure $\mathrm{A}$, a line segment drawn from the center of the circle perpendicular to one of the sides is called an apothem of the polygon. (FIGURE CAN'T COPY) (a) Show that the length of the apothem in Figure A is $\sqrt{3} / 2$ (b) Show that the length of one side of the heptagon in Figure $B$ is $2 \sin \left(180^{\circ} / 7\right)$ (c) Use a calculator to evaluate the expressions in parts (a) and (b). Round each answer to four decimal places, and note how close the two values are. Approximately two thousand years ago, Heron of Alexandria made use of this coincidence when he used the length of the apothem of the hexagon to approximate the length of the side of the heptagon. (The apothem of the hexagon can be constructed with ruler and compass; the side of the regular heptagon cannot.)

Figure A shows a regular hexagon inscribed in a circle of radius $1 .$ Figure $\mathrm{B}$ shows a regular heptagon (seven-sided polygon) inscribed in a circle of radius $1 .$ In Figure $\mathrm{A}$, a line segment drawn from the center of the circle perpendicular to one of the sides is called an apothem of the polygon.
(FIGURE CAN'T COPY)
(a) Show that the length of the apothem in Figure A is $\sqrt{3} / 2$
(b) Show that the length of one side of the heptagon in Figure $B$ is $2 \sin \left(180^{\circ} / 7\right)$
(c) Use a calculator to evaluate the expressions in parts (a) and (b). Round each answer to four decimal places, and note how close the two values are. Approximately two thousand years ago, Heron of Alexandria made use of this coincidence when he used the length of the apothem of the hexagon to approximate the length of the side of the heptagon. (The apothem of the hexagon can be constructed with ruler and compass; the side of the regular heptagon cannot.)

Key Concepts

Regular Inscribed Polygons

Regular inscribed polygons are figures with all sides and angles equal, placed in such a way that every vertex lies on the circumference of a circle. This property allows the use of central angles and trigonometric relationships to determine various geometric measures such as side lengths, apothems, and areas.

Apothem

The apothem of a regular polygon is the perpendicular distance from its center to any of its sides. It plays an essential role in connecting circle geometry to polygon geometry and is useful for calculating areas and relating the side lengths of the polygon to the radius of the circumscribed circle.

Chord Length in a Circle

In a circle, a chord is a line segment with both endpoints on the circle. For a regular polygon inscribed in a circle, each side is a chord, and its length can be determined using the formula 2R sin(?/2), where R is the radius of the circle and ? is the central angle subtended by the chord. This relationship is fundamental for deriving side lengths of inscribed regular polygons.

Trigonometric Relationships in Polygons

Trigonometric functions such as sine and cosine bridge the gap between linear measurements and angles in polygons. These relationships enable the derivation of formulas for key geometric quantities like apothems and chord lengths. By analyzing the central and inscribed angles of the polygon, one can systematically deduce various parameters critical to both theoretical and practical geometric problems.

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