Area of Regular Polygons

A complete guide to regular polygons: area formulas, relationships between side length, inscribed and circumscribed radii, and an interactive visualization tool. Also try the Regular Polygon Calculator.

Area Formulas for Regular Polygons

A regular polygon has all sides equal and all interior angles equal. Below is a regular pentagon (5 sides):

Where:

• $x$ = side length
• $r$ = radius of inscribed circle (apothem)
• $R$ = radius of circumscribed circle
• $n$ = number of sides
• $t = \frac{360^\circ}{n}$ = central angle

Relationships Between $x$, $r$, and $R$

From right triangle trigonometry: $$ \tan\left(\frac{t}{2}\right) = \frac{x/2}{r} $$ $$ \sin\left(\frac{t}{2}\right) = \frac{x/2}{R} $$ Solving for $r$ and $R$: $$ r = \frac{x}{2} \cot\left(\frac{180^\circ}{n}\right) $$ $$ R = \frac{x}{2} \csc\left(\frac{180^\circ}{n}\right) $$

Formula 1: Using Side Length and Apothem

Area of one triangle = $\frac{1}{2} \cdot x \cdot r$
Total area: $$ A = \frac{1}{2} n x r $$

Formula 2: Using Circumradius $R$

Area of one triangle = $\frac{1}{2} R^2 \sin(t)$
Total area: $$ A = \frac{1}{2} n R^2 \sin\left(\frac{360^\circ}{n}\right) $$

Formula 3: Using Side Length Only

Substitute $r$ from above into Formula 1: $$ A = \frac{1}{4} n x^2 \cot\left(\frac{180^\circ}{n}\right) $$

Formula 4: Using Apothem Only

From $x = 2r \tan(180^\circ/n)$ and Formula 1: $$ A = n r^2 \tan\left(\frac{180^\circ}{n}\right) $$

Polygon Names by Number of Sides

Interactive Visualization

Instructions:

1. Click "Draw" to generate the polygon.
2. Adjust the number of sides ($n$) using the slider or input box.
3. Observe how the polygon approximates a circle as $n$ increases.

Solutions to Exercises

Problem: As $n$ increases, what happens to the areas of the circumscribed circle, the polygon, and the inscribed circle?

Solution:

• The circumscribed circle area remains constant ($\pi R^2$).
• The inscribed circle area increases and approaches the circumscribed circle area. Its radius is $r = R \cos(180^\circ/n) \to R$ as $n \to \infty$.
• The polygon area increases and approaches the circumscribed circle area from below. As $n \to \infty$, the polygon becomes a circle.

Thus, all three areas converge to the same value: $\pi R^2$.

More Geometry Resources

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