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
| Sides | Name |
|---|---|
| 3 | Equilateral Triangle |
| 4 | Square |
| 5 | Pentagon |
| 6 | Hexagon |
| 7 | Heptagon |
| 8 | Octagon |
| 9 | Nonagon |
| 10 | Decagon |
| 11 | Undecagon |
| 12 | Dodecagon |
Interactive Visualization
Instructions:
- Click "Draw" to generate the polygon.
- Adjust the number of sides (\(n\)) using the slider or input box.
- 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: \(\pi R^2\).