Arc, Sector, Segment & Chord

1. In the previous chapter, you learned how right triangles help you calculate lengths and angles you cannot measure directly (Pythagoras and trig).
2. Now we’ll use those same skills inside circles: chords create triangles, and sectors create curved lengths and “slice” areas.

Circle Vocabulary That Makes Geometry Precise

Arc Length

1. A full circle is $360^{\circ}$ and has circumference $2 \pi r$.
2. So an arc is the same fraction of the circumference as its angle is of $360^{\circ}$ : $$\ell=\frac{\theta}{360^{\circ}} \times 2 \pi r$$
3. Useful rearrangement (solving for $\theta$ ): $$\theta=\frac{180 \ell}{\pi r}$$
4. And solving for $r$ : $$r=\frac{180 \ell}{\pi \theta}$$

Sector Area

1. A full circle has area: $$A=\pi r^2$$
2. A sector is a fraction of the full area:
   1. If $\theta$ in degrees: $$A_{\text {sector }}=\frac{\theta}{360} \times \pi r^2$$
   2. If $\theta$ in radians: $$A_{\text {sector }}=\frac{1}{2} r^2 \theta$$

Perimeter of a Sector

1. A sector’s perimeter includes:
   1. two radii, each length $r$
   2. one arc, length $s$
2. So: $$P_{\text {sector }}=2 r+s$$

Chords Create Isosceles and Right Triangles

1. Two radii and a chord form an isosceles triangle (because all radii are equal).
2. If you draw a line from the center to the midpoint of the chord, it:
   1. bisects the chord (splits it into two equal halves)
   2. is perpendicular to the chord
3. This creates two right-angled triangles.
4. Once you have right triangles, you can use:
   1. Pythagoras, or
   2. trigonometry (especially sine)

Finding the Length of a Chord

1. Let:
   1. radius $r$
   2. central angle $\theta$ (in degrees)
   3. chord length $c$
2. Using the two right triangles formed by splitting the isosceles triangle: $$c=2 r \sin \left(\frac{\theta}{2}\right)$$
3. To find the angle when you know the chord: $$\theta=2 \arcsin \left(\frac{c}{2 r}\right)$$