Regular polygons and composite figures are key players in geometry. They show up everywhere, from simple shapes to complex designs. Knowing how to calculate their areas is super useful in real-life situations.

Area formulas for regular polygons use apothems and perimeters. For irregular shapes, we break them down into simpler parts. These methods help us tackle all sorts of polygon problems, from basic to advanced.

Area Formulas for Regular Polygons

Apothem and perimeter for polygon area

• Calculate the area of a regular polygon using the formula $A = \frac{1}{2}ap$
  • $a$ represents the apothem, the perpendicular distance from the center of the polygon to any side (radius of the inscribed circle)
  • $p$ represents the perimeter of the polygon, the total distance around the polygon
• Find the perimeter of a regular polygon with $n$ sides of length $s$ using the formula $p = ns$
  • For example, a regular hexagon with side length 5 cm has a perimeter of $p = 6 \times 5 = 30$ cm
• Combine the area and perimeter formulas to express the area as $A = \frac{1}{2}ans$
  • This formula is useful when given the number of sides and side length of a regular polygon

Trigonometric ratios in regular polygons

• Calculate the central angle $\theta$ in a regular polygon with $n$ sides using the formula $\theta = \frac{360^\circ}{n}$
  • For instance, in a regular pentagon ($n = 5$), the central angle is $\theta = \frac{360^\circ}{5} = 72^\circ$
• Determine the apothem $a$ using the trigonometric formula $a = \frac{s}{2\tan(\frac{\theta}{2})}$
  • $s$ represents the side length of the polygon
  • $\theta$ represents the central angle
  • This formula is helpful when the side length and number of sides are known
• Calculate the side length $s$ using the trigonometric formula $s = 2a\tan(\frac{\theta}{2})$
  • This formula is useful when the apothem and number of sides are given

Area of Composite and Irregular Figures

Composite figures and total area

• Identify the basic shapes that compose the composite figure (squares, rectangles, triangles, circles)
• Calculate the area of each individual shape using their respective area formulas
  • Rectangle: $A = lw$ (length $\times$ width)
  • Triangle: $A = \frac{1}{2}bh$ (base $\times$ height)
  • Circle: $A = \pi r^2$ (radius squared $\times \pi$)
• Add the areas of all the individual shapes to find the total area of the composite figure
  • For example, a composite figure consisting of a rectangle (6 cm $\times$ 4 cm) and a triangle (base 4 cm, height 3 cm) has a total area of $24 + 6 = 30$ cm²

Irregular polygons and area decomposition

• Divide the irregular polygon into triangles and quadrilaterals by drawing diagonals from one vertex to another
  • Choose diagonals that create familiar shapes like right triangles or rectangles
• Calculate the area of each triangle using the formula $A = \frac{1}{2}bh$
  • $b$ represents the base of the triangle
  • $h$ represents the height of the triangle
• Calculate the area of each quadrilateral using their respective area formulas
  • Rectangle: $A = lw$
  • Parallelogram: $A = bh$
  • Trapezoid: $A = \frac{1}{2}(b_1 + b_2)h$ (average of parallel sides $\times$ height)
• Add the areas of all the triangles and quadrilaterals to find the total area of the irregular polygon
  • For instance, an irregular pentagon divided into two triangles (areas 12 cm² and 15 cm²) and a trapezoid (area 28 cm²) has a total area of $12 + 15 + 28 = 55$ cm²