🔷Honors Geometry Unit 11 – Areas of Polygons and Circles

Key Concepts and Definitions

• Polygon a closed plane figure with at least three straight sides and angles
• Regular polygon has all sides of equal length and all angles of equal measure
• Apothem perpendicular distance from the center of a regular polygon to any side
• Circumference distance around the outside of a circle, calculated by $2\pi r$
• Area the space inside a two-dimensional figure, measured in square units
  • Calculated using specific formulas depending on the shape
• Pi ($\pi$) mathematical constant approximately equal to 3.14159
  • Represents the ratio of a circle's circumference to its diameter

Types of Polygons

• Triangle a polygon with three sides and three angles (equilateral, isosceles, scalene)
• Quadrilateral a polygon with four sides and four angles
  • Includes squares, rectangles, parallelograms, trapezoids, and rhombuses
• Pentagon a polygon with five sides and five angles
• Hexagon a polygon with six sides and six angles
• Heptagon a polygon with seven sides and seven angles
• Octagon a polygon with eight sides and eight angles
• Decagon a polygon with ten sides and ten angles

Area Formulas for Basic Polygons

• Triangle area $A = \frac{1}{2}bh$, where $b$ is the base and $h$ is the height
• Rectangle area $A = lw$, where $l$ is the length and $w$ is the width
• Square area $A = s^2$, where $s$ is the side length
• Parallelogram area $A = bh$, where $b$ is the base and $h$ is the height
• Trapezoid area $A = \frac{1}{2}(b_1 + b_2)h$, where $b_1$ and $b_2$ are the parallel sides and $h$ is the height
• Regular polygon area $A = \frac{1}{2}ap$, where $a$ is the apothem and $p$ is the perimeter

Circles and Their Components

• Circle a closed plane curve with all points equidistant from the center
• Radius a line segment from the center of a circle to any point on the circle
  • Denoted by the variable $r$
• Diameter a line segment passing through the center of a circle, with endpoints on the circle
  • Equal to twice the radius, denoted by the variable $d$
• Chord a line segment connecting any two points on a circle
• Tangent a line that intersects a circle at exactly one point
• Secant a line that intersects a circle at two points
• Arc a portion of the circumference of a circle
  • Measured in degrees or radians

Area of Circles and Circular Sectors

• Circle area $A = \pi r^2$, where $r$ is the radius
• Circular sector area $A = \frac{1}{2}r^2\theta$, where $r$ is the radius and $\theta$ is the central angle in radians
  • To find the area using degrees, use the formula $A = \frac{\theta}{360}\pi r^2$
• Relationship between radians and degrees $\theta_{radians} = \frac{\theta_{degrees}}{180}\pi$
• One radian approximately equal to 57.3 degrees

Advanced Polygon Area Calculations

• Kite area $A = \frac{1}{2}d_1d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals
• Rhombus area $A = \frac{1}{2}d_1d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals
  • Can also be calculated using the base and height formula for a parallelogram
• Area of a polygon by triangulation divide the polygon into triangles and sum their areas
• Heron's formula for triangle area $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$ (semi-perimeter) and $a$, $b$, and $c$ are side lengths
• Area of a polygon using coordinates $A = \frac{1}{2}|(x_1y_2 + x_2y_3 + ... + x_ny_1) - (y_1x_2 + y_2x_3 + ... + y_nx_1)|$

Real-World Applications

• Architecture calculating areas for flooring, roofing, and landscaping
• Engineering designing parts and components with specific area requirements
• Graphic design determining the area of shapes in logos and layouts
• Land surveying measuring the area of plots and parcels
• Cartography calculating the area of regions on maps
• Textile industry determining the amount of fabric needed for clothing and upholstery
• Agriculture calculating the area of fields for planting and irrigation

Common Mistakes and How to Avoid Them

• Confusing diameter and radius in circle calculations
  • Remember that the diameter is twice the radius
• Forgetting to divide by 2 in the triangle area formula
  • The formula is $A = \frac{1}{2}bh$, not $A = bh$
• Using the wrong formula for a specific polygon
  • Identify the polygon type and use the appropriate formula
• Misidentifying the height of a polygon
  • The height is the perpendicular distance from the base to the opposite vertex or side
• Incorrectly measuring or calculating the apothem of a regular polygon
  • The apothem is the perpendicular distance from the center to any side
• Mixing up units (e.g., using inches instead of feet)
  • Ensure all measurements are in the same units before calculating the area
• Rounding prematurely in multi-step calculations
  • Carry out all calculations with full precision and round the final answer if necessary