The Main Idea

The length of a circular arc tells us how far along the circle’s edge we travel when sweeping out an angle. The key is that the angle must be measured in radians, because radians directly connect an angle to the arc length. If [latex]\theta[/latex] is the central angle in radians and [latex]r[/latex] is the circle’s radius, then the arc length is given by [latex]s = r\theta[/latex]. This formula works because radians are defined as arc length divided by radius.

Quick Tips: Finding Arc Length

1. Use the formula: [latex]s = r\theta[/latex], where [latex]s[/latex] is arc length, [latex]r[/latex] is radius, and [latex]\theta[/latex] is in radians.
2. Covert first if needed: If the angle is given in degrees, convert to radians before using the formula.
3. Check the units: The arc length will be in the same unit as the radius (e.g., if [latex]r[/latex] is in cm, [latex]s[/latex] will be in cm).
4. Fraction of the circle: You can also think of arc length as a fraction of the whole circumference:
   1. [latex]s = \dfrac{\theta}{2\pi}\cdot (2\pi r)[/latex].
5. Real-world connection: Arc length is like the “distance walked” along the edge of the circle — useful for wheels, gears, and circular tracks.

The Main Idea

A sector of a circle is like a slice of pizza — it’s the region between two radii and the arc that connects them. The area of that slice depends on how big the angle is and how large the circle is. If the radius is [latex]r[/latex] and the central angle is [latex]\theta[/latex] in radians, then the area of the sector is given by [latex]A = \tfrac{1}{2}r^2\theta[/latex]. This formula works because radians directly connect angle size to the fraction of the circle’s area.

Quick Tips: Finding Area of a Sector

1. Use the formula: [latex]A = \tfrac{1}{2}r^2\theta[/latex], where [latex]\theta[/latex] is in radians.
2. Convert if necessary: If the angle is in degrees, convert to radians before plugging it in.
3. Check the units: The area will be in square units (e.g., [latex]\text{cm}^2[/latex]) if the radius is in cm.
4. Fraction of the whole circle: The formula also comes from [latex]\dfrac{\theta}{2\pi}\cdot \pi r^2[/latex]; the fraction of the circle’s total area.
5. Think pizza or pie: A small angle gives a skinny slice, a big angle gives a bigger slice — the formula measures the “size of the slice.”

The Main Idea

When an object moves along a circular path, we can describe its motion in two connected ways: angular speed and linear speed. Angular speed measures how fast the angle changes (in radians per unit of time), while linear speed measures how fast the distance along the arc changes (in distance per unit of time). The two are tied together by the radius:
[latex]v = r\omega[/latex], where [latex]v[/latex] is linear speed, [latex]r[/latex] is the radius, and [latex]\omega[/latex] is angular speed. This relationship lets us move between “spinning speed” (angular) and “traveling speed” (linear).

Quick Tips: Finding Area of a Sector

1. Linear speed formula: [latex]\omega = \dfrac{\theta}{t}[/latex], where [latex]\theta[/latex] is in radians and [latex]t[/latex] is time.
2. Angular speed formula: [latex]v = \dfrac{s}{t}[/latex], where [latex]s[/latex] is the arc length traveled.
3. Connect them: Use [latex]s = r\theta[/latex] to link the two formulas, giving [latex]v = r\omega[/latex].
4. Units Matter:
   1. Linear speed: feet per second, meters per second, etc.
   2. Angular speed: radians per second (or per minute).
5. Think Real-World: On a spinning wheel, all points share the same angular speed, but points farther from the center move faster linearly.