Circles: Circumference and Area
Radius and Diameter
A circle is the set of all points a fixed distance from a center point. That fixed distance is the radius (r). The diameter (d) is a segment passing through the center connecting two points on the circle — it's twice the radius:
d = 2rCircumference
The circumference is the distance around the circle — its "perimeter." It's related to the diameter by the constant \pi (pi), approximately 3.14159:
C = \pi d = 2\pi rArea of a Circle
The area enclosed by a circle of radius r is:
A = \pi r^2Arcs and Sectors
An arc is a portion of the circle's circumference, and a sector is the pie-slice-shaped region bounded by two radii and the arc between them. Both are measured as a fraction of the whole circle, based on the central angle \theta (in degrees) that the arc/sector spans out of the full 360^{\circ}.
Arc Length
\text{Arc length} = \frac{\theta}{360^{\circ}} \times 2\pi rSector Area
\text{Sector area} = \frac{\theta}{360^{\circ}} \times \pi r^2In both cases, you're just taking the fraction of the circle that the central angle represents, then scaling the full circumference or full area by that same fraction.
Example
A circle has radius 6. Find its circumference, its area, and the area of a sector with a central angle of 60^{\circ}.
- Circumference: C = 2\pi r = 2\pi(6) = 12\pi
- Full circle area: A = \pi r^2 = \pi (6)^2 = 36\pi
- The sector's central angle is 60^{\circ}, which is \frac{60}{360} = \frac{1}{6} of the full circle.
- Sector area: \frac{1}{6}\times 36\pi = 6\pi
Key Takeaways
- d = 2r
- Circumference: C = 2\pi r = \pi d
- Area: A = \pi r^2
- Arc length and sector area both scale by the fraction \frac{\theta}{360^{\circ}} of the whole circle.