Question

Architecture A bird's-eye view of the Pentagon reveals five similar pentagons. Each side of the outside pentagon is about 920 feet. Each side of the innermost pentagon is about 360 feet. a. Find the scale factor of the outside pentagon to the innermost pentagon. b. Find the perimeter of the outside pentagon. c. Find the perimeter of the innermost pentagon. d. Find the ratio of the perimeter of the outside pentagon to the perimeter of the innermost pentagon. e. Tell how the ratio in part \(d\) compares to the scale factor of the pentagons.

Short answer

a: 23/9, b: 4600 feet, c: 1800 feet, d: 23/9, e: Same as scale factor.

Step-by-step solution

Understand the problem

We need to find the scale factor and perimeters for the pentagons and compare ratios. The outer pentagon has side lengths of 920 feet, and the innermost pentagon's side lengths are 360 feet.

Calculate the scale factor

The scale factor is the ratio of the side length of the outside pentagon to the side length of the innermost pentagon. Thus, the scale factor is \( \frac{920}{360} \). Simplifying this fraction yields \( \frac{23}{9} \).

Find the perimeter of the outside pentagon

The perimeter of a pentagon is calculated by summing the lengths of all its sides. For the outside pentagon, multiply the side length by 5: \( 5 \times 920 = 4600 \) feet.

Find the perimeter of the innermost pentagon

Similarly, for the innermost pentagon, multiply its side length by 5: \( 5 \times 360 = 1800 \) feet.

Calculate the ratio of the perimeters

The ratio of the perimeter of the outside pentagon to the innermost pentagon is \( \frac{4600}{1800} \). Simplifying this fraction gives \( \frac{23}{9} \).

Compare the ratio to the scale factor

The ratio of the perimeters \( \frac{23}{9} \) is equal to the scale factor \( \frac{23}{9} \). This shows that the perimeter ratio is the same as the scale factor.

Key concepts

Scale Factor

In a geometric context, a scale factor is a ratio that represents the relationship between two similar geometric figures. It tells us how much one shape has been magnified or reduced in size compared to another.
To find the scale factor for the Pentagon problem, we compare the side length of the outside pentagon (920 feet) with the innermost pentagon (360 feet). We do this by forming a fraction: \[ \text{Scale Factor} = \frac{\text{Outside Side Length}}{\text{Innermost Side Length}} = \frac{920}{360} \]Upon simplifying this fraction, we get \( \frac{23}{9} \). This shows that each side of the outside pentagon is \( \frac{23}{9} \) times larger than the corresponding side of the innermost pentagon.
This equivalency is crucial in understanding that if the scale factor is greater than 1, the outside pentagon is larger than the innermost pentagon. Conversely, a scale factor less than 1 would indicate a reduction in size.

Perimeter Calculation

Perimeter calculation is a fundamental concept in geometry that involves summing the lengths of all sides of a polygon. For a pentagon, which has five sides, this is particularly straightforward.
Starting with the outside pentagon, each side is 920 feet in length. To find its perimeter, multiply the side length by 5:\[ \text{Perimeter of Outside Pentagon} = 5 \times 920 = 4600 \text{ feet} \]The same method is used for the innermost pentagon, with each side being 360 feet:\[ \text{Perimeter of Innermost Pentagon} = 5 \times 360 = 1800 \text{ feet} \]These calculations demonstrate how the total length around any regular polygon, regardless of its size, can be easily determined using its side lengths. Understanding perimeter calculation is key to solving many geometry problems.

Ratio Comparison

Comparing ratios allows us to understand the relationship between different geometric measurements, such as the perimeter of two similar shapes.
In our pentagon example, we compare the perimeter of the outside pentagon to that of the innermost pentagon. This ratio is calculated by dividing their perimeters:\[ \text{Perimeter Ratio} = \frac{\text{Perimeter of Outside Pentagon}}{\text{Perimeter of Innermost Pentagon}} = \frac{4600}{1800} \]Simplifying gives \( \frac{23}{9} \), the same as the scale factor. This shows that not only the side lengths but also the overall sizes (as measured by perimeter) have increased or decreased by this ratio.
Recognizing ratio equivalency helps confirm the similarity of shapes, demonstrating that even if the shapes look significantly different in size, their proportions and relationships remain consistent. This consistency is a foundation for deeper understanding in geometric analysis.