Understanding the Regular Hexagon and Triangle AOB

A regular hexagon is a six-sided polygon where all sides are equal in length, and all interior angles are equal. The question specifies a regular hexagon ABCDEF with a side length of 36 cm. The point O refers to the center of the hexagon. Triangle AOB is formed by connecting the center O to two adjacent vertices, A and B.

Properties of a Regular Hexagon's Center

When you connect the center of a regular hexagon to each of its vertices, the hexagon is divided into six congruent triangles. For a regular hexagon, these six triangles are not just congruent but are also equilateral triangles.

This means that:

• Triangle AOB, formed by the center O and adjacent vertices A and B, is an equilateral triangle.
• All sides of triangle AOB are equal in length.
• The sides OA, OB, and AB are all equal to the side length of the hexagon.

Given that the side of the hexagon is 36 cm, the side length of the equilateral triangle AOB is also 36 cm.

Calculating the Area of an Equilateral Triangle

The area of an equilateral triangle can be calculated using the formula:

Area $= \frac{\sqrt{3}}{4} \times \text{(side)}^2$

In our case, the side length of triangle AOB is 36 cm.

Step-by-Step Area Calculation

1. Identify the shape of triangle AOB: It's an equilateral triangle because ABCDEF is a regular hexagon and O is its center.
2. Determine the side length of triangle AOB: It is equal to the side length of the hexagon, which is 36 cm.
3. Use the formula for the area of an equilateral triangle: Area $= \frac{\sqrt{3}}{4} s^2$.
4. Substitute the side length ($s = 36$ cm) into the formula.

Area of triangle AOB $= \frac{\sqrt{3}}{4} \times (36 \, \text{cm})^2$

Area $= \frac{\sqrt{3}}{4} \times (36 \times 36) \, \text{cm}^2$

Area $= \frac{\sqrt{3}}{4} \times 1296 \, \text{cm}^2$

Now, divide 1296 by 4:

$1296 \div 4 = 324$

So, the Area of triangle AOB $= \sqrt{3} \times 324 \, \text{cm}^2$

Area $= 324\sqrt{3} \, \text{cm}^2$

The area of the triangle AOB is $324\sqrt{3}$ cm².

Revision Table: Regular Hexagon Geometry

Additional Information: Area Formulas and Properties

Understanding basic geometric shapes and their properties is crucial for solving geometry problems. Here are some related concepts:

• Interior Angle of a Regular Hexagon: Each interior angle of a regular hexagon is $(6-2) \times 180 / 6 = 4 \times 180 / 6 = 720 / 6 = 120$ degrees.
• Central Angle: The angle formed by connecting the center to two adjacent vertices (like $\angle$ AOB) is $360 / 6 = 60$ degrees. Since OA=OB (radii of the circumcircle) and $\angle$ AOB = 60 degrees, triangle AOB must be equilateral.
• Area of Regular Hexagon: The area of a regular hexagon can be found by summing the areas of the six equilateral triangles, or using the formula: Area $= \frac{3\sqrt{3}}{2} s^2$, where 's' is the side length. For this hexagon, the total area would be $6 \times (324\sqrt{3}) = 1944\sqrt{3}$ cm².
• Other Triangle Area Formulas:
  • Base and Height: Area $= \frac{1}{2} \times \text{base} \times \text{height}$
  • Using Sine: Area $= \frac{1}{2} ab \sin(C)$ (where a, b are side lengths and C is the included angle)

Applying the correct properties of regular polygons, like the fact that connecting the center to vertices of a regular hexagon creates equilateral triangles, simplifies area calculations significantly.