The Base Of A Solid Right Pyramid Is A Regular Hexagon With A Radius Of 2 X 2x 2 X Units And An Apothem Of X 3 X\sqrt{3} X 3 ​ Units.Which Expression Represents The Area Of The Base Of The Pyramid?A. X 2 3 X^2\sqrt{3} X 2 3 ​ Units 2 ^2 2 B.

Understanding the Problem

The problem presents a solid right pyramid with a regular hexagon as its base. The radius of the hexagon is given as $2x$ units, and the apothem is given as $x\sqrt{3}$ units. We are tasked with finding the expression that represents the area of the base of the pyramid.

Properties of a Regular Hexagon

A regular hexagon is a six-sided polygon with all sides and angles equal. It can be divided into six equilateral triangles, each sharing a common vertex. The apothem of a regular hexagon is the distance from the center of the hexagon to one of its vertices, which is also the height of each equilateral triangle.

Calculating the Area of the Base

To find the area of the base of the pyramid, we need to calculate the area of the regular hexagon. Since the hexagon can be divided into six equilateral triangles, we can find the area of one triangle and then multiply it by 6 to get the total area of the hexagon.

Finding the Area of an Equilateral Triangle

The area of an equilateral triangle with side length $s$ is given by the formula:

$$A = \frac{\sqrt{3}}{4} s^2$$

In this case, the side length of each equilateral triangle is equal to the radius of the hexagon, which is $2x$ units. Therefore, the area of one equilateral triangle is:

$$A = \frac{\sqrt{3}}{4} (2x)^2$$

Simplifying the Expression

Simplifying the expression, we get:

$$A = \frac{\sqrt{3}}{4} (4x^2)$$

$$A = \sqrt{3} x^2$$

Finding the Total Area of the Hexagon

Since the hexagon can be divided into six equilateral triangles, the total area of the hexagon is six times the area of one triangle:

$$A_{hexagon} = 6 \times A$$

$$A_{hexagon} = 6 \times \sqrt{3} x^2$$

$$A_{hexagon} = 6\sqrt{3} x^2$$

Conclusion

The expression that represents the area of the base of the pyramid is $6\sqrt{3} x^2$ units$^2$. This is the correct answer.

Final Answer

The final answer is: $\boxed{6\sqrt{3} x^2}$

Understanding the Problem

The problem presents a solid right pyramid with a regular hexagon as its base. The radius of the hexagon is given as $2x$ units, and the apothem is given as $x\sqrt{3}$ units. We are tasked with finding the expression that represents the area of the base of the pyramid.

Q&A

Q: What is the formula for the area of an equilateral triangle?

A: The formula for the area of an equilateral triangle with side length $s$ is given by:

$$A = \frac{\sqrt{3}}{4} s^2$$

Q: How do we find the area of the base of the pyramid?

A: To find the area of the base of the pyramid, we need to calculate the area of the regular hexagon. Since the hexagon can be divided into six equilateral triangles, we can find the area of one triangle and then multiply it by 6 to get the total area of the hexagon.

Q: What is the relationship between the side length of an equilateral triangle and the radius of the hexagon?

A: The side length of each equilateral triangle is equal to the radius of the hexagon, which is $2x$ units.

Q: How do we simplify the expression for the area of one equilateral triangle?

A: Simplifying the expression, we get:

$$A = \frac{\sqrt{3}}{4} (2x)^2$$

$$A = \frac{\sqrt{3}}{4} (4x^2)$$

$$A = \sqrt{3} x^2$$

Q: What is the total area of the hexagon?

A: Since the hexagon can be divided into six equilateral triangles, the total area of the hexagon is six times the area of one triangle:

$$A_{hexagon} = 6 \times A$$

$$A_{hexagon} = 6 \times \sqrt{3} x^2$$

$$A_{hexagon} = 6\sqrt{3} x^2$$

Q: What is the final answer?

A: The expression that represents the area of the base of the pyramid is $6\sqrt{3} x^2$ units$^2$.

Common Mistakes

• Not understanding the properties of a regular hexagon
• Not knowing the formula for the area of an equilateral triangle
• Not simplifying the expression for the area of one equilateral triangle
• Not multiplying the area of one triangle by 6 to get the total area of the hexagon

Tips and Tricks

• Make sure to understand the properties of a regular hexagon before starting the problem
• Use the formula for the area of an equilateral triangle to find the area of one triangle
• Simplify the expression for the area of one triangle before multiplying it by 6
• Double-check your work to make sure you get the correct answer

Conclusion

The base of a solid right pyramid with a regular hexagon as its base can be found using the formula for the area of an equilateral triangle and the properties of a regular hexagon. By following the steps outlined in this article, you can find the expression that represents the area of the base of the pyramid.