A rope between two spokes

Inscribe a regular polygon with $n\ge3$ sides in a circle of radius $R$. Connect two neighboring vertices to the center O. The triangle formed has two sides equal to $R$ and a base equal to the side $\ell$ of the polygon.

The $n$ equal central angles complete one turn. Each measures $360°/n$. Draw the height of this triangle from O. For symmetry, it cuts the base and the central angle in half.

Demonstration of the relationship with the sine

In one of the halves a right-angled triangle appears. Its angle at O ​​measures $180°/n$, its hypotenuse is $R$ and the opposite side is $\ell/2$.

We define the sine of an acute angle as the ratio of the opposite side to the hypotenuse. Therefore:

$$\sin\frac{180°}{n}=\frac{\ell/2}{R}=\frac{\ell}{2R}.$$

Insulating side length:

$$\ell=2R\sin\frac{180°}{n}.$$

On the circumference of radius 1, the relationship reduces to $\sin(180°/n)=\ell/2$. In high school, the same angular measurement will be written as $\pi/n$ radians.

Application to the hexagon

A regular hexagon can be divided, by its radii, into six equilateral triangles: each central angle measures $360°/6=60°$. Thus, the side of the hexagon measures $R$.

Replacing $n=6$ and $\ell=R$:

$$\sin30°=\frac{R}{2R}=\frac12.$$

Sine has no unit: it is a ratio between lengths measured in the same unit. Doubling the radius doubles the side, but does not change the quotient.

In everyday life

  • Hex nuts. The head of a screw is a regular hexagon; the central half-angle sine connects the side to the radius.
  • STOP Sign. The sign is a regular octagon: eight central angles of $45^\circ$.
  • Spoke wheels. On a bicycle wheel, the distance between the ends of two neighboring spokes is a chord, calculated with the sine of half an angle.
Everyday polygons: with 8 sides, the STOP sign; with 6, the hexagonal nut. The string is $2r\operatorname{sen}\frac{\theta}{2}$.

Exercises with resolution

1. A regular inscribed hexagon has radius 7 cm. Calculate its side and perimeter.

Check resolution 1

The side measures 7 cm. The perimeter is $6\cdot7=42$ cm.

2. An inscribed square has side $R\sqrt2$. Use the chord relationship to find $\sin45°$.

Check resolution 2

With $n=4$, $\sin45°=\ell/(2R)=R\sqrt2/(2R)=\sqrt2/2$.

In the interactive book

The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:

This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.

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