Understanding Similar Square Pyramids

Oct 13, 2025 by miodeya.com 29 views

Introduction

In geometry, similar figures are those that have the same shape but not necessarily the same size. When dealing with similar figures, we often need to find the ratio of their corresponding dimensions, such as side lengths or areas. In this article, we will explore the concept of similar square pyramids and determine the fraction that represents the ratio of the base area of the smaller pyramid to the base area of the larger pyramid.

Understanding Similar Square Pyramids

A square pyramid is a three-dimensional shape with a square base and four triangular faces that meet at the apex. When two square pyramids are similar, it means that their corresponding sides are proportional. In other words, if the side length of the smaller pyramid is $\frac{3}{4}$ the side length of the larger pyramid, then the ratio of their corresponding sides is $\frac{3}{4}$ .

The Formula for the Base Area of a Square Pyramid

The base area of a square pyramid is given by the formula:

A = s^2

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

Finding the Ratio of Base Areas

Let's assume that the side length of the smaller pyramid is $s_1$ and the side length of the larger pyramid is $s_2$ . We are given that $s_1 = \frac{3}{4}s_2$ . We need to find the ratio of the base areas of the two pyramids, which is given by:

\frac{A_1}{A_2} = \frac{s_1^2}{s_2^2}

Substituting the value of $s_1$ in terms of $s_2$ , we get:

\frac{A_1}{A_2} = \frac{\left(\frac{3}{4}s_2\right)^2}{s_2^2}

Simplifying the expression, we get:

\frac{A_1}{A_2} = \frac{\frac{9}{16}s_2^2}{s_2^2}

Canceling out the common factor of $s_2^2$ , we get:

\frac{A_1}{A_2} = \frac{9}{16}

Conclusion

In conclusion, the ratio of the base area of the smaller pyramid to the base area of the larger pyramid is $\frac{9}{16}$ . This means that if the base area of the larger pyramid is 16 units, then the base area of the smaller pyramid is 9 units.

Example

Suppose we have two square pyramids with side lengths 4 and 6 units, respectively. The ratio of their corresponding sides is $\frac{4}{6} = \frac{2}{3}$ . Using the formula for the base area, we can calculate the base areas of the two pyramids:

A_1 = 4^2 = 16

A_2 = 6^2 = 36

The ratio of the base areas is:

\frac{A_1}{A_2} = \frac{16}{36} = \frac{4}{9}

However, this is not the ratio we are looking for. We need to find the ratio of the base areas of the two pyramids, which is given by:

\frac{A_1}{A_2} = \frac{\left(\frac{3}{4}s_2\right)^2}{s_2^2}

Substituting the value of $s_2 = 6$ , we get:

\frac{A_1}{A_2} = \frac{\left(\frac{3}{4}\cdot 6\right)^2}{6^2}

Simplifying the expression, we get:

\frac{A_1}{A_2} = \frac{\left(\frac{9}{2}\right)^2}{6^2}

\frac{A_1}{A_2} = \frac{\frac{81}{4}}{36}

\frac{A_1}{A_2} = \frac{81}{4} \cdot \frac{1}{36}

\frac{A_1}{A_2} = \frac{9}{16}

Therefore, the ratio of the base area of the smaller pyramid to the base area of the larger pyramid is $\frac{9}{16}$ .

Final Answer

Q&A: Similar Square Pyramids

Q: What is the definition of similar figures?

A: Similar figures are those that have the same shape but not necessarily the same size. In other words, if two figures have the same shape, but one is larger or smaller than the other, they are similar.

Q: What is the formula for the base area of a square pyramid?

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

A = s^2

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

Q: How do we find the ratio of base areas of two similar square pyramids?

A: To find the ratio of base areas of two similar square pyramids, we need to find the ratio of their corresponding side lengths. If the side length of the smaller pyramid is $\frac{3}{4}$ the side length of the larger pyramid, then the ratio of their corresponding side lengths is $\frac{3}{4}$ . We can then use this ratio to find the ratio of their base areas.

Q: What is the ratio of base areas of two similar square pyramids if the side length of the smaller pyramid is $\frac{3}{4}$ the side length of the larger pyramid?

A: The ratio of base areas of two similar square pyramids is given by:

\frac{A_1}{A_2} = \frac{s_1^2}{s_2^2}

Substituting the value of $s_1$ in terms of $s_2$ , we get:

\frac{A_1}{A_2} = \frac{\left(\frac{3}{4}s_2\right)^2}{s_2^2}

Simplifying the expression, we get:

\frac{A_1}{A_2} = \frac{\frac{9}{16}s_2^2}{s_2^2}

Canceling out the common factor of $s_2^2$ , we get:

\frac{A_1}{A_2} = \frac{9}{16}

Q: Can we use the ratio of base areas to find the side length of the smaller pyramid?

A: Yes, we can use the ratio of base areas to find the side length of the smaller pyramid. If we know the ratio of base areas and the side length of the larger pyramid, we can use the formula:

\frac{A_1}{A_2} = \frac{s_1^2}{s_2^2}

to find the side length of the smaller pyramid.

Q: What is the relationship between the side length of the smaller pyramid and the side length of the larger pyramid?

A: The side length of the smaller pyramid is $\frac{3}{4}$ the side length of the larger pyramid.

Q: Can we use the ratio of base areas to find the volume of the smaller pyramid?

A: Yes, we can use the ratio of base areas to find the volume of the smaller pyramid. If we know the ratio of base areas and the volume of the larger pyramid, we can use the formula:

\frac{V_1}{V_2} = \frac{A_1}{A_2}

to find the volume of the smaller pyramid.

Q: What is the relationship between the volume of the smaller pyramid and the volume of the larger pyramid?

A: The volume of the smaller pyramid is $\frac{27}{64}$ the volume of the larger pyramid.

Final Answer

The final answer is $\boxed{\frac{9}{16}}$ .