A Pyramid Has A Square Base With Side $s$. The Height Of The Pyramid Is $\frac{2}{3}$ That Of Its Side. What Is The Expression For The Volume Of The Pyramid?A. $V=2s^2$ B. $V=2s^3$ C. $V=\frac{2}{3}s^2$

In this article, we will delve into the world of geometry and explore the concept of a pyramid's volume. Specifically, we will examine a pyramid with a square base and determine the expression for its volume based on the given information.

The Basics of a Pyramid

A pyramid is a three-dimensional shape with a square or triangular base and sides that meet at the apex. The base of the pyramid is a square with side length $s$. The height of the pyramid is given as $\frac{2}{3}$ that of its side.

Calculating the Volume of a Pyramid

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

$$V = \frac{1}{3}Bh$$

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

Determining the Area of the Base

The base of the pyramid is a square with side length $s$. The area of a square is given by:

$$A = s^2$$

Therefore, the area of the base is $s^2$.

Determining the Height of the Pyramid

The height of the pyramid is given as $\frac{2}{3}$ that of its side. Since the side length is $s$, the height can be expressed as:

$$h = \frac{2}{3}s$$

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:

$$V = \frac{1}{3}Bh$$

Substituting the values for $B$ and $h$, we get:

$$V = \frac{1}{3}s^2 \times \frac{2}{3}s$$

Simplifying the expression, we get:

$$V = \frac{2}{9}s^3$$

Conclusion

In this article, we have determined the expression for the volume of a pyramid with a square base and side length $s$. The height of the pyramid is $\frac{2}{3}$ that of its side. Using the formula for the volume of a pyramid, we have calculated the expression for the volume as $\frac{2}{9}s^3$.

Answer

The correct answer is:

• $V = \frac{2}{9}s^3$

Comparison with Other Options

Let's compare our answer with the other options:

• $V = 2s^2$ is incorrect because it does not take into account the height of the pyramid.
• $V = 2s^3$ is incorrect because it does not take into account the height of the pyramid and the area of the base.
• $V = \frac{2}{3}s^2$ is incorrect because it does not take into account the height of the pyramid.

Final Thoughts

$$$$

In our previous article, we explored the concept of a pyramid's volume and determined the expression for its volume based on the given information. In this article, we will address some common questions and concerns related to the topic.

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

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

$$V = \frac{1}{3}Bh$$

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 the base of a pyramid?

A: The area of the base of a pyramid is given by:

$$A = s^2$$

where $s$ is the side length of the base.

Q: What is the height of the pyramid in the given problem?

A: The height of the pyramid is given as $\frac{2}{3}$ that of its side. Since the side length is $s$, the height can be expressed as:

$$h = \frac{2}{3}s$$

Q: How do I calculate the volume of a pyramid with a triangular base?

A: The formula for the volume of a pyramid with a triangular base is the same as for a square base:

$$V = \frac{1}{3}Bh$$

However, the area of the base ($B$) will be different. For a triangular base, the area is given by:

$$B = \frac{1}{2}bh$$

where $b$ is the base length and $h$ is the height of the triangle.

Q: Can I use the formula for the volume of a pyramid to calculate the height of the pyramid?

A: Yes, you can use the formula for the volume of a pyramid to calculate the height of the pyramid. Rearranging the formula to solve for $h$, we get:

$$h = \frac{3V}{B}$$

Q: What are some real-world applications of the concept of a pyramid's volume?

A: The concept of a pyramid's volume has many real-world applications, including:

• Architecture: The volume of a pyramid is an important consideration in the design of buildings and monuments.
• Engineering: The volume of a pyramid is used in the design of storage containers and other structures.
• Geology: The volume of a pyramid is used to calculate the volume of rock formations and other geological features.

Q: Can I use the formula for the volume of a pyramid to calculate the volume of a cone?

A: Yes, you can use the formula for the volume of a pyramid to calculate the volume of a cone. A cone is a type of pyramid with a circular base. The formula for the volume of a cone is:

$$V = \frac{1}{3}\pi r^2h$$

where $r$ is the radius of the base and $h$ is the height of the cone.

Conclusion

In this article, we have addressed some common questions and concerns related to the concept of a pyramid's volume. We hope this article has provided a clear understanding of the topic and has been helpful in answering your questions.