The area of a regular hexagon of side ‘a’ is equal to
(3√3/2)a 2square units
Understanding the Area of a Regular Hexagon
A regular hexagon is a polygon with six equal sides and six equal interior angles. To find the area of a regular hexagon, we can divide it into simpler shapes.
A key property of a regular hexagon is that it can be perfectly divided into six congruent equilateral triangles by drawing lines from the center of the hexagon to each vertex. If the side length of the regular hexagon is 'a', then each of these equilateral triangles also has a side length of 'a'.
The area of an equilateral triangle with side length 's' is given by the formula:
\[ \text{Area of equilateral triangle} = \frac{\sqrt{3}}{4} s^2 \]In our case, the side length 's' is equal to the side length of the hexagon, which is 'a'. So, the area of one equilateral triangle is:
\[ \text{Area of one triangle} = \frac{\sqrt{3}}{4} a^2 \]Since the regular hexagon is made up of exactly six of these identical equilateral triangles, the total area of the hexagon is 6 times the area of one triangle.
\[ \text{Area of regular hexagon} = 6 \times (\text{Area of one triangle}) \] \[ \text{Area of regular hexagon} = 6 \times \left( \frac{\sqrt{3}}{4} a^2 \right) \]Now, we simplify the expression:
\[ \text{Area of regular hexagon} = \frac{6 \sqrt{3}}{4} a^2 \]By cancelling the common factor of 2 in the numerator and denominator, we get:
\[ \text{Area of regular hexagon} = \frac{3 \sqrt{3}}{2} a^2 \]So, the area of a regular hexagon of side 'a' is \( \frac{3\sqrt{3}}{2}a^2 \) square units.
Comparing with the Options
Let's compare our calculated area with the given options:
- Option 1: \( (\frac{\sqrt{2}}{3})a^2 \) square units
- Option 2: \( (\frac{3\sqrt{3}}{2})a^2 \) square units
- Option 3: \( (\frac{1}{3})a^2 \) square units
- Option 4: \( (\frac{\sqrt{3}}{2})a^2 \) square units
Our calculated area, \( (\frac{3\sqrt{3}}{2})a^2 \), matches Option 2.
Revision Table: Area of Regular Hexagon and Related Formulas
| Concept | Formula (side 'a') | Notes |
|---|---|---|
| Area of Equilateral Triangle | \( \frac{\sqrt{3}}{4} a^2 \) | Building block for regular hexagon area |
| Area of Regular Hexagon | \( \frac{3\sqrt{3}}{2} a^2 \) | Sum of areas of 6 equilateral triangles |
| Perimeter of Regular Hexagon | \( 6a \) | Sum of 6 equal sides |
| Interior Angle of Regular Hexagon | \( 120^\circ \) | \( \frac{(n-2) \times 180^\circ}{n} \) with n=6 |
Additional Information on Regular Hexagons
Regular hexagons have many interesting properties:
- Symmetry: They have both rotational and reflectional symmetry.
- Angles: Each interior angle is \( 120^\circ \), and each exterior angle is \( 60^\circ \).
- Circumradius (R): The distance from the center to any vertex is equal to the side length 'a'. So, R = a.
- Inradius or Apothem (r): The distance from the center to the midpoint of any side is the height of one of the equilateral triangles. This is \( \frac{\sqrt{3}}{2} a \). So, \( r = \frac{\sqrt{3}}{2} a \).
- Packing: Regular hexagons can tile a plane without any gaps or overlaps, which is why they are often seen in nature (like honeycomb) and in design.
Knowing these properties can be helpful in solving various geometry problems involving regular hexagons.
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