11.2 Areas of regular polygons and composite figures

Area Formulas for Regular Polygons

The area of any regular polygon can be calculated with a single formula, as long as you know two measurements: the apothem and the perimeter. This section also covers how trigonometry connects those measurements when you're only given partial information.

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Apothem and perimeter for polygon area

The core formula for the area of a regular polygon is:

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

• $a$ is the apothem: the perpendicular distance from the center of the polygon to the midpoint of any side. You can also think of it as the radius of the circle inscribed inside the polygon.
• $p$ is the perimeter: the total distance around the polygon.

This formula works because you can divide any regular polygon into $n$ congruent isosceles triangles, each with a base equal to one side and a height equal to the apothem. The area of each triangle is $\frac{1}{2} \cdot s \cdot a$, and there are $n$ of them, so the total area is $\frac{1}{2} \cdot a \cdot (n \cdot s) = \frac{1}{2}ap$.

For a regular polygon with $n$ sides of length $s$, the perimeter is simply:

$p = ns$

So you can also write the area as $A = \frac{1}{2}ans$.

Trigonometric ratios in regular polygons

When you're given the side length but not the apothem (or vice versa), trig ratios fill in the gap.

Step 1: Find the central angle. Each of the $n$ congruent triangles formed from the center has a central angle of:

$\theta = \frac{360°}{n}$

For a regular pentagon ($n = 5$), that's $\theta = \frac{360°}{5} = 72°$.

Step 2: Find the apothem from the side length. The apothem bisects the central angle and meets the midpoint of a side, creating a right triangle. Using tangent:

$a = \frac{s}{2\tan\!\left(\frac{\theta}{2}\right)}$

For that pentagon with $s = 8$ cm: $a = \frac{8}{2\tan(36°)} = \frac{8}{2(0.7265)} \approx 5.51$ cm.

Step 3 (alternate): Find the side length from the apothem. If you know $a$ instead, rearrange:

$s = 2a\tan\!\left(\frac{\theta}{2}\right)$

Once you have both $a$ and $s$, plug them into $A = \frac{1}{2}ap$ to get the area.

Area of Composite and Irregular Figures

Composite and irregular figures aren't covered by a single formula. Instead, you break them into simpler shapes whose areas you already know how to find, then combine the results.

Composite figures and total area

A composite figure is made up of two or more basic shapes joined together (or one shape with a piece removed). Here's the general approach:

1. Identify the basic shapes that make up the figure: rectangles, triangles, semicircles, trapezoids, etc.

2. Calculate each area using the appropriate formula:

   • Rectangle: $A = lw$
   • Triangle: $A = \frac{1}{2}bh$
   • Circle: $A = \pi r^2$ (use $\frac{1}{2}\pi r^2$ for a semicircle)

3. Add areas of shapes that are joined together. Subtract areas of shapes that are cut out or removed.

Watch for overlapping regions. If two shapes share an area, you'll double-count it unless you subtract the overlap.

Irregular polygons and area decomposition

An irregular polygon has sides and angles that aren't all congruent, so there's no single formula. The strategy is decomposition: split it into familiar shapes.

1. Draw diagonals from one vertex to divide the polygon into triangles and quadrilaterals. Pick diagonals that create right triangles or rectangles when possible, since those are easiest to work with.

2. Calculate each sub-area using the appropriate formula:

   • Triangle: $A = \frac{1}{2}bh$
   • Parallelogram: $A = bh$
   • Trapezoid: $A = \frac{1}{2}(b_1 + b_2)h$

3. Sum all the sub-areas to get the total.

If coordinates are given, you can also use the coordinate geometry (shoelace) formula, but for this course, decomposition into basic shapes is the standard method.