The length of the base edge of a pyramid with a regular hexagon base is represented as [tex]$x$[/tex]. The height of the pyramid is [tex]$3$[/tex] times longer than the base edge.
The height of the pyramid can be represented as $\square$.
The area of an equilateral triangle with length [tex]$x$[/tex] is [tex]$\frac{x^2 \sqrt{3}}{4}$[/tex] units [tex]$^2$[/tex].
The area of the hexagon base is $\square$ times the area of the equilateral triangle.
The volume of the pyramid is $\square$ [tex]$x^3 \sqrt{3}$[/tex] units [tex]$^3$[/tex].
Answer (1)
The height of the pyramid is 3 x .
The area of the hexagon base is 6 times the area of an equilateral triangle with side x .
The volume of the pyramid is 2 3 x 3 3 units 3 .
Therefore, the final answers are: 3 x , 6 , 2 3
Explanation
Problem Introduction and Setup Let's break down this pyramid problem step by step. We're given a pyramid with a regular hexagon as its base. The length of each side of the hexagon is x , and the height of the pyramid is 3 x . We need to find the height of the pyramid in terms of x , how many times greater the area of the hexagon is compared to an equilateral triangle with side x , and the volume of the pyramid.
Finding the Height of the Pyramid The problem states that the height of the pyramid is 3 times the length of the base edge, which is x . Therefore, the height of the pyramid is simply 3 x .
Calculating the Area of the Hexagon A regular hexagon can be divided into 6 equilateral triangles. The area of one equilateral triangle with side length x is given as 4 x 2 3 . Since the hexagon is made up of 6 such triangles, the area of the hexagon is 6 × 4 x 2 3 = 4 6 x 2 3 = 2 3 x 2 3 .
Comparing Areas: Hexagon vs. Equilateral Triangle Now, we want to find how many times greater the area of the hexagon is compared to the area of one equilateral triangle. To do this, we divide the area of the hexagon by the area of the equilateral triangle: 4 x 2 3 2 3 x 2 3 = 2 3 x 2 3 × x 2 3 4 = 2 x 2 3 12 x 2 3 = 6 So, the area of the hexagon is 6 times the area of the equilateral triangle.
Calculating the Volume of the Pyramid The volume V of a pyramid is given by the formula V = 3 1 B h , where B is the area of the base and h is the height. In our case, the base is the hexagon with area B = 2 3 x 2 3 , and the height is h = 3 x . Plugging these values into the formula, we get: V = 3 1 × 2 3 x 2 3 × 3 x = 6 9 x 3 3 = 2 3 x 3 3 = 1.5 x 3 3
Final Answer Therefore, the height of the pyramid is 3 x , the area of the hexagon is 6 times the area of the equilateral triangle, and the volume of the pyramid is 2 3 x 3 3 .
Examples
Understanding pyramids and their volumes is useful in architecture and engineering. For example, when designing a building with a pyramid-shaped roof, architects need to calculate the volume to estimate the materials required. Knowing the relationship between the base edge, height, and volume allows for efficient planning and cost estimation. Also, the concept of dividing a hexagon into equilateral triangles is used in tessellations and pattern design, which can be applied in creating visually appealing structures and designs.
