Circumference and Arc Length

Big Picture

The circumference and arc of a a circle can be found with the appropriate theorems and formulas.

Key Terms

Circumference: The distance around a circle.

Sector (of a circle): The area bounded by two radii and the arc between the endpoints of the radii.

Segment (of a circle): The area of a circle that is bounded by a chord and the arc with the same endpoints as the chord.

Parts of a Circle

• Circle: Set of all points in a plane that are a given distance from another point (the center).
• Circumference: Perimeter of a circle.
• Radius: Any segment from the center to a point on the circle (written as r).
• Diameter: Any segment from one point on the circle through the center to another point on the circle (written as d).
• d = 2r

Circumference

The circumference of a circle is just the perimeter of a circle.

• Circumeference = $$2πr = πd$$
• π (an irrational number pronounced “pi”) = $$\frac{c}{d} = \frac{circumference}{diameter} \approx 3.14159265358979323846...$$

Area

Theorem: The area of a circle is  times the square of the radius.

• Area = $$\pi r^2$$

Arc Length

Arcs are fractional portions of circles. They are measured in degree measures and linear measures.

• Degree measure: fractional part of a 360° complete circle that the arc is in
• Linear measure: length of the arc

Corollary: The ratio of the arc length to the circumference is equal to the ratio of the arc measure to 360°.

• length of AB = $$\frac{m\overset{ \huge\frown}{AB}}{360^{\circ}} . 2 \pi r = \frac{m\overset{ \huge\frown}{AB}}{360^{\circ}} . \pi d$$

Circumference and Arc Length Cont.

Sectors

The sector is a fractional part of the area of the circle, often written as ($$\frac{m}{360^{\circ}}$$)fraction of the circle.

Finding the area of the sector is like finding the fractional part of the area of the circle.

Theorem: Area of a sector = $$\frac{m\overset{ \huge\frown}{AB}}{360^{\circ}} . \pi r^2$$

Basically, the fractional portion of the circle that the sector  represents is multiplied by the total area of the circle, giving the area of the circle.

If the arc length (s) is known, $$A_{sector} = (\frac{1}{2}sr)$$.

Segments

The area of the segment is the area of the sector minus the area of the triangle made by the radii.

Do not confuse the segment of a circle with line segment!