1. Understanding Area of Regular Polygons
A regular polygon is a polygon with all sides equal and all angles equal. Because of this perfect symmetry, the polygon can be split into identical triangles. This idea helps us calculate its area.
For basic cases like equilateral triangles, squares, and regular hexagons, we use formulas based on their symmetry and simple geometry.
1.1. Idea of Dividing into Congruent Triangles
A regular polygon with \(n\) sides can be divided into \(n\) congruent triangles by joining each vertex to the centre of the polygon.
Each triangle has:
- Central angle = \(\dfrac{360^\circ}{n}\)
- Two equal sides (radii of the circumcircle)
This method is especially useful in understanding the area formula.
2. Area of a Square (Regular Quadrilateral)
A square is the simplest regular polygon with 4 equal sides and 4 right angles.
2.1. Formula
If the side length of a square is \(a\), then:
\( \text{Area} = a^2 \)
2.2. Example
If a square has side \(a = 6 \,\text{cm}\):
\( \text{Area} = 6^2 = 36 \,\text{cm}^2 \)
3. Area of an Equilateral Triangle
An equilateral triangle is a regular polygon with 3 equal sides and each interior angle equal to \(60^\circ\).
3.1. Formula
If each side is \(a\), then the area is:
\( \displaystyle \text{Area} = \dfrac{\sqrt{3}}{4}a^2 \)
3.2. Explanation (Simple)
This formula is obtained by dropping an altitude, forming two 30-60-90 right triangles. Using the height \(\dfrac{\sqrt{3}}{2}a\), the final area becomes:
\( \dfrac{1}{2} \times a \times \dfrac{\sqrt{3}}{2}a = \dfrac{\sqrt{3}}{4}a^2 \)
3.3. Example
If \(a = 4\):
\( \text{Area} = \dfrac{\sqrt{3}}{4} \times 16 = 4\sqrt{3} \)
4. Area of a Regular Hexagon
A regular hexagon is especially interesting because it can be divided into 6 equilateral triangles of equal area.
4.1. Using the Division Into Triangles
If the side length of a regular hexagon is \(a\), then the area is:
- Area of one equilateral triangle: \(\dfrac{\sqrt{3}}{4}a^2\)
- There are 6 such triangles.
\( \text{Area}_{\text{hexagon}} = 6 \times \dfrac{\sqrt{3}}{4}a^2 = \dfrac{3\sqrt{3}}{2}a^2 \)
4.2. Example
If \(a = 5\):
\( \text{Area} = \dfrac{3\sqrt{3}}{2} \times 25 = \dfrac{75\sqrt{3}}{2} \)
5. General Area Formula Using Apothem (Optional Understanding)
For any regular polygon, we can use the apothem (a perpendicular from center to any side) to find the area. This is helpful when the number of sides is large.
5.1. Formula
If the polygon has perimeter \(P\) and apothem \(a\):
\( \text{Area} = \dfrac{1}{2}Pa \)
5.2. Why This Works
Because the polygon can be divided into \(n\) congruent triangles, each with base equal to the side and height equal to the apothem.