Geometry Honors Notes: Areas of Circles and Sectors

Polygons

  • Polygon: Union of segments in a plane intersecting only at endpoints, with no consecutive segments collinear.
  • Perimeter: Sum of side lengths.
  • Diagonal: Segment joining nonconsecutive vertices.
  • The following formulas are used for an n-gon:
    • Sum of Interior Angles: (n−2)(180)(n-2)(180)
    • Sum of Exterior Angles: 360360
    • Number of Diagonals: n(n−3)2\frac{n(n-3)}{2}

Regular Polygons

  • Regular Polygon: Convex polygon, both equilateral and equiangular.
  • Apothem: Distance from center to a side.
  • Radius: Distance from center to a vertex.
  • Area of Regular Polygon: A=12aPA = \frac{1}{2} aP, where a is the apothem and P is the perimeter.

Circumference of a Circle

  • Circumference: Distance around the edge of the circle.
  • C=2πrC = 2 \pi r or C=πdC = \pi d

Area of Circles

  • Area: A=πr2A = \pi r^2
  • Annulus: Region between two concentric circles; area is the difference between the areas of the circles.
  • For similar figures, the ratio of their linear units squared is equal to the ratio of their areas.

Arc Length and Sector Area

  • Arc Length: Distance along the circle between two endpoints.
  • Arc Length Formula: x360⋅2πr\frac{x}{360} \cdot 2 \pi r, where x is the arc's degree measure.
  • Sector Area: Region bounded by two radii and an arc.
  • Sector Area Formula: A=x360⋅πr2A = \frac{x}{360} \cdot \pi r^2
  • Area of Segment: Area of Sector - Area of Triangle

Maximizing Area

  • Involves maximizing or minimizing geometric figures' dimensions, area, or volume.
  • To find the maximum/minimum of a quadratic equation, find the vertex.
  • The vertex of a parabola is the point (h,k)(h, k), where h=−b2ah = \frac{-b}{2a} and k=f(h)k = f(h).

Geometric Probability

  • Deals with probability problems involving geometric concepts like area.
  • P(event) = \frac{\text{# of favorable outcomes}}{\text{total # of outcomes}}