Areas of Polygons and Circles

Understanding how to calculate the area of various geometric shapes is a fundamental skill in geometry. This guide covers the essential area formulas for polygons, circles, and methods for solving complex composite figures.

Basic Polygon Area Formulas

Here are the standard area formulas for common polygons:

• Triangle: $A = \frac{1}{2}bh$ (where $b$ is the base and $h$ is the height)
• Parallelogram: $A = bh$
• Trapezoid: $A = \frac{1}{2}(b_1 + b_2)h$ (where $b_1$ and $b_2$ are the parallel bases)
• Rhombus and Kite: $A = \frac{1}{2}d_1 d_2$ (where $d_1$ and $d_2$ are the lengths of the diagonals)

Heron's Formula for Triangles

If you know the lengths of all three sides of a triangle ($a$, $b$, and $c$) but not the height, you can use Heron's Formula. First, calculate the semi-perimeter ($s$):

$s = \frac{a + b + c}{2}$

Then, find the area ($A$):

$A = \sqrt{s(s - a)(s - b)(s - c)}$

Area of Regular Polygons

A regular polygon has equal side lengths and equal interior angles. The general formula for the area of a regular polygon is:

$A = \frac{1}{2}aP$

where $a$ is the apothem (the distance from the center to the midpoint of a side) and $P$ is the perimeter.

Example: Find the area of a regular hexagon with side length 6. A regular hexagon can be divided into 6 identical equilateral triangles. The area of one equilateral triangle with side length $s$ is $\frac{\sqrt{3}}{4}s^2$.

1. Area of one triangle: $A_{tri} = \frac{\sqrt{3}}{4}(6^2) = \frac{\sqrt{3}}{4}(36) = 9\sqrt{3}$
2. Total area of the hexagon: $A = 6 \times 9\sqrt{3} = 54\sqrt{3}$

Area of Circles

The area of a circle depends entirely on its radius ($r$):

$A = \pi r^2$

(Note: If you are given the diameter, simply divide it by 2 to find the radius before calculating the area.)

Composite Figures and Shaded Regions

Composite figures are complex shapes made up of two or more simple shapes. To find the area of a composite figure, you can either add the areas of the simple shapes together or subtract an empty region from a larger shape.

Example: Find the area of the shaded region between a square with side 10 and an inscribed circle.

1. Find the area of the square: The side length is 10. $A_{square} = 10^2 = 100$
2. Find the area of the circle: Because the circle is inscribed, its diameter equals the side length of the square (10). Therefore, the radius is $r = 5$. $A_{circle} = \pi (5^2) = 25\pi$
3. Subtract to find the shaded region: The shaded area is the square's area minus the circle's area. $A_{shaded} = 100 - 25\pi$ (Approximated as $100 - 78.54 = 21.46$)