A Solid Oblique Pyramid Has A Regular Hexagonal Base With An Area Of $54 \sqrt{3} , \text{cm}^2$ And An Edge Length Of 6 Cm.Angle BAC Measures $60^{\circ}$.What Is The Volume Of The Pyramid?A. $72 \sqrt{3} ,

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Introduction

In geometry, a pyramid is a three-dimensional shape with a base and sides that converge at the apex. A solid oblique pyramid is a type of pyramid where the base is not a square or rectangle, but rather a regular hexagon. In this article, we will explore the properties of a solid oblique pyramid with a regular hexagonal base and calculate its volume.

Properties of a Regular Hexagonal Base

A regular hexagon is a six-sided polygon with all sides and angles equal. The area of a regular hexagon can be calculated using the formula:

A=332a2A = \frac{3\sqrt{3}}{2}a^2

where AA is the area and aa is the length of one side.

Given that the area of the regular hexagonal base is $54 \sqrt{3} , \text{cm}^2$, we can use the formula to find the length of one side:

543=332a254 \sqrt{3} = \frac{3\sqrt{3}}{2}a^2

Solving for aa, we get:

a2=543Γ—233a^2 = \frac{54 \sqrt{3} \times 2}{3\sqrt{3}}

a2=36a^2 = 36

a=6 cma = 6 \, \text{cm}

Properties of a Solid Oblique Pyramid

A solid oblique pyramid is a type of pyramid where the base is not a square or rectangle, but rather a regular hexagon. The height of the pyramid is the distance from the apex to the base. In this case, we are given that angle BAC measures $60^{\circ}$.

Calculating the Volume of the Pyramid

The volume of a pyramid can be calculated using the formula:

V=13BhV = \frac{1}{3}Bh

where VV is the volume, BB is the area of the base, and hh is the height.

We already know the area of the base, which is $54 \sqrt{3} , \text{cm}^2$. However, we need to find the height of the pyramid.

Finding the Height of the Pyramid

To find the height of the pyramid, we can use the properties of a 30-60-90 triangle. Since angle BAC measures $60^{\circ}$, we know that triangle ABC is a 30-60-90 triangle.

In a 30-60-90 triangle, the ratio of the sides is 1:√3:2. Since the length of side AC is 6 cm, we can use this ratio to find the height of the pyramid.

Let hh be the height of the pyramid. Then, we can set up the following proportion:

h6=32\frac{h}{6} = \frac{\sqrt{3}}{2}

Solving for hh, we get:

h=6Γ—32h = 6 \times \frac{\sqrt{3}}{2}

h=33 cmh = 3\sqrt{3} \, \text{cm}

Calculating the Volume of the Pyramid

Now that we have the area of the base and the height of the pyramid, we can calculate the volume using the formula:

V=13BhV = \frac{1}{3}Bh

V=13Γ—543Γ—33V = \frac{1}{3} \times 54 \sqrt{3} \times 3\sqrt{3}

V=54Γ—3Γ—3V = 54 \times \sqrt{3} \times \sqrt{3}

V=54Γ—3V = 54 \times 3

V=162 cm3V = 162 \, \text{cm}^3

However, we need to consider the given answer choices to determine if our answer is correct.

Discussion

The given answer choices are:

A. $72 \sqrt{3} , \text{cm}^3$

B. $162 , \text{cm}^3$

C. $216 , \text{cm}^3$

D. $243 , \text{cm}^3$

Our calculated answer is $162 , \text{cm}^3$, which is option B.

Conclusion

In this article, we explored the properties of a solid oblique pyramid with a regular hexagonal base and calculated its volume. We used the formula for the area of a regular hexagon to find the length of one side, and then used the properties of a 30-60-90 triangle to find the height of the pyramid. Finally, we calculated the volume using the formula for the volume of a pyramid. Our calculated answer is $162 , \text{cm}^3$, which is option B.

References

  • [1] Geometry: A Comprehensive Introduction
  • [2] Calculus: Early Transcendentals
  • [3] Mathematics for Engineers and Scientists

Note: The references provided are for general information purposes only and are not specific to the problem at hand.

Introduction

In our previous article, we explored the properties of a solid oblique pyramid with a regular hexagonal base and calculated its volume. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q1: What is the formula for the area of a regular hexagon?

A1: The formula for the area of a regular hexagon is:

A=332a2A = \frac{3\sqrt{3}}{2}a^2

where AA is the area and aa is the length of one side.

Q2: How do you find the height of a solid oblique pyramid?

A2: To find the height of a solid oblique pyramid, you can use the properties of a 30-60-90 triangle. Since angle BAC measures $60^{\circ}$, we know that triangle ABC is a 30-60-90 triangle. In a 30-60-90 triangle, the ratio of the sides is 1:√3:2. Since the length of side AC is 6 cm, we can use this ratio to find the height of the pyramid.

Q3: What is the formula for the volume of a pyramid?

A3: The formula for the volume of a pyramid is:

V=13BhV = \frac{1}{3}Bh

where VV is the volume, BB is the area of the base, and hh is the height.

Q4: How do you calculate the volume of a solid oblique pyramid with a regular hexagonal base?

A4: To calculate the volume of a solid oblique pyramid with a regular hexagonal base, you need to find the area of the base and the height of the pyramid. Then, you can use the formula for the volume of a pyramid:

V=13BhV = \frac{1}{3}Bh

Q5: What is the relationship between the area of the base and the height of a solid oblique pyramid?

A5: The area of the base and the height of a solid oblique pyramid are related by the formula for the volume of a pyramid:

V=13BhV = \frac{1}{3}Bh

This means that if you know the area of the base and the height of the pyramid, you can calculate the volume.

Q6: Can you provide an example of a solid oblique pyramid with a regular hexagonal base?

A6: Yes, consider a solid oblique pyramid with a regular hexagonal base and an edge length of 6 cm. The area of the base is $54 \sqrt{3} , \text{cm}^2$. Using the formula for the area of a regular hexagon, we can find the length of one side:

543=332a254 \sqrt{3} = \frac{3\sqrt{3}}{2}a^2

Solving for aa, we get:

a2=36a^2 = 36

a=6 cma = 6 \, \text{cm}

The height of the pyramid is 3√3 cm. Using the formula for the volume of a pyramid, we can calculate the volume:

V=13BhV = \frac{1}{3}Bh

V=13Γ—543Γ—33V = \frac{1}{3} \times 54 \sqrt{3} \times 3\sqrt{3}

V=54Γ—3Γ—3V = 54 \times \sqrt{3} \times \sqrt{3}

V=54Γ—3V = 54 \times 3

V=162 cm3V = 162 \, \text{cm}^3

Conclusion

In this article, we answered some frequently asked questions related to the topic of a solid oblique pyramid with a regular hexagonal base. We provided formulas and examples to help illustrate the concepts. We hope this article has been helpful in understanding the properties of a solid oblique pyramid with a regular hexagonal base.

References

  • [1] Geometry: A Comprehensive Introduction
  • [2] Calculus: Early Transcendentals
  • [3] Mathematics for Engineers and Scientists

Note: The references provided are for general information purposes only and are not specific to the problem at hand.