A Solid Right Pyramid Has A Regular Hexagonal Base With An Area Of $5.2 , \text{cm}^2$ And A Height Of H Cm H \, \text{cm} H Cm $.Which Expression Represents The Volume Of The Pyramid?A. 1 5 ( 5.2 ) H Cm 3 \frac{1}{5}(5.2)h \, \text{cm}^3 5 1 ( 5.2 ) H Cm 3 B.

Mar 11, 2025 by ADMIN 265 views

A Solid Right Pyramid with a Regular Hexagonal Base: Finding the Volume

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Introduction

A solid right pyramid is a three-dimensional shape with a polygonal base and triangular faces that meet at the apex. In this case, we are given a pyramid with a regular hexagonal base, which means all sides of the base are equal in length. The area of the base is given as $5.2 \, \text{cm}^2$ , and the height of the pyramid is $h \, \text{cm}$ . Our goal is to find the expression that represents the volume of the pyramid.

Understanding the Formula for the Volume of a Pyramid

The formula for the volume of a pyramid is given by:

V = \frac{1}{3}Bh

where $V$ is the volume, $B$ is the area of the base, and $h$ is the height of the pyramid. This formula is derived from the fact that the pyramid can be divided into three congruent pyramids, each with a triangular base and a height equal to one-third of the original height.

Finding the Area of the Regular Hexagonal Base

A regular hexagon can be divided into six equilateral triangles, each with a side length equal to the side length of the hexagon. The area of an equilateral triangle with side length $s$ is given by:

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

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

B = 6 \times \frac{\sqrt{3}}{4}s^2 = \frac{3\sqrt{3}}{2}s^2

Expressing the Volume of the Pyramid

Now that we have the formula for the area of the regular hexagonal base, we can substitute this expression into the formula for the volume of the pyramid:

V = \frac{1}{3}Bh = \frac{1}{3} \times \frac{3\sqrt{3}}{2}s^2 \times h = \frac{\sqrt{3}}{2}s^2h

However, we are given that the area of the base is $5.2 \, \text{cm}^2$ , so we can substitute this value into the expression for the volume:

V = \frac{\sqrt{3}}{2}s^2h = \frac{\sqrt{3}}{2} \times \left(\frac{2 \times 5.2}{3\sqrt{3}}\right)^2h = \frac{1}{3} \times 5.2 \times h = \frac{1}{3} \times 5.2h

Conclusion

In conclusion, the expression that represents the volume of the pyramid is $\frac{1}{3} \times 5.2h \, \text{cm}^3$ . This expression is derived from the formula for the volume of a pyramid and the given area of the regular hexagonal base.

Discussion

The discussion category for this problem is mathematics, specifically geometry and trigonometry. The problem requires the application of formulas and theorems from these subjects to find the expression that represents the volume of the pyramid.

Final Answer

The final answer is $\boxed{\frac{1}{3} \times 5.2h \, \text{cm}^3}$ .

Introduction

In our previous article, we discussed the problem of finding the expression that represents the volume of a solid right pyramid with a regular hexagonal base. We derived the formula for the volume of the pyramid and applied it to the given area of the base. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A

Q: What is the formula for the volume of a pyramid?

A: The formula for the volume of a pyramid is given by:

V = \frac{1}{3}Bh

where $V$ is the volume, $B$ is the area of the base, and $h$ is the height of the pyramid.

Q: What is the area of a regular hexagonal base?

A: The area of a regular hexagonal base can be found by dividing the hexagon into six equilateral triangles. The area of an equilateral triangle with side length $s$ is given by:

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

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

B = 6 \times \frac{\sqrt{3}}{4}s^2 = \frac{3\sqrt{3}}{2}s^2

Q: How do I find the side length of the regular hexagonal base?

A: To find the side length of the regular hexagonal base, you can use the given area of the base and the formula for the area of a regular hexagon:

B = \frac{3\sqrt{3}}{2}s^2

Rearranging this formula to solve for $s$ , we get:

s = \sqrt{\frac{2B}{3\sqrt{3}}}

Q: What is the relationship between the area of the base and the side length of the regular hexagonal base?

A: The area of the base is directly proportional to the square of the side length of the regular hexagonal base. This means that if the side length of the base increases, the area of the base will also increase.

Q: Can I use this formula to find the volume of any pyramid?

A: Yes, you can use this formula to find the volume of any pyramid, as long as you know the area of the base and the height of the pyramid.

Q: What are some real-world applications of finding the volume of a pyramid?

A: Finding the volume of a pyramid has many real-world applications, such as:

Calculating the volume of a container or a storage tank
Determining the amount of material needed for construction
Estimating the weight of a pyramid or a structure
Calculating the volume of a pyramid-shaped object, such as a tomb or a monument

Conclusion

In conclusion, finding the volume of a pyramid with a regular hexagonal base requires a good understanding of geometry and trigonometry. By using the formula for the volume of a pyramid and the given area of the base, we can find the expression that represents the volume of the pyramid. We hope that this Q&A section has helped to clarify any doubts and provide additional information on the topic.

Final Answer

The final answer is $\boxed{\frac{1}{3} \times 5.2h \, \text{cm}^3}$ .