Question

Determine whether the following statement is true or false. Explain. If the lengths of the sides of a regular polygon are doubled, then its area is also doubled.

Short answer

The statement is false; the area becomes four times larger, not double.

Step-by-step solution

Understanding the Problem

We need to determine whether doubling the length of the sides of a regular polygon results in doubling its area.

Recall Area Formula

The area of a regular polygon with side length \( s \) and number of sides \( n \) can be expressed as \( A = \frac{1}{4}ns^2 \cot\left(\frac{\pi}{n}\right) \). This formula incorporates both the side length and a trigonometric factor based on the number of sides.

Modify Side Length

To model the change stated in the problem, if the side length \( s \) is doubled to become \( 2s \), the new area \( A' \) becomes \( A' = \frac{1}{4}n(2s)^2 \cot\left(\frac{\pi}{n}\right) \).

Simplify New Area Expression

Substituting \( 2s \) in the area formula gives \( A' = \frac{1}{4}n(4s^2) \cot\left(\frac{\pi}{n}\right) = 4 \cdot \frac{1}{4}ns^2 \cot\left(\frac{\pi}{n}\right) = 4A \).

Final Conclusion

The new area \( A' \) is 4 times the original area \( A \), not double the original area. Thus, the statement in the problem is false.

Key concepts

Understanding Regular Polygons

A regular polygon is a shape with all sides and angles equal. Examples include squares, equilateral triangles, and regular hexagons. These polygons are simple yet hold a lot of symmetry.
Understanding this symmetry is essential as it impacts how formulas, like those for area and perimeter, apply to all regular polygons. For any regular polygon, the uniformity hugely simplifies calculations since each set of sides and angles behaves the same way.
Regular polygons come in a range with three or more sides. The number of sides, often represented as \( n \), dictates how the polygon looks and behaves spatially. This is important when considering the geometric and trigonometric relationships that define these shapes.

Exploring the Area Formula of Regular Polygons

To find the area of a regular polygon, a specific formula is used which incorporates both the side length and the number of sides. The formula is:
\[A = \frac{1}{4}ns^2 \cot\left(\frac{\pi}{n}\right)\]where:

• \( A \) is the area of the polygon,
• \( n \) is the number of sides,
• \( s \) is the length of one side,
• \( \cot\left(\frac{\pi}{n}\right) \) arises from trigonometry, accounting for angle computations within the polygon.

This formula shows how influential side length \( s \) and number of sides \( n \) are in determining the polygon's area. The presence of trigonometry through \( \cot \) connects the size and shape by considering internal angles.
This relationship underscores why the polygon's area changes in a non-linear way when its side length changes.

Trigonometric Relationships in the Area Formula

Trigonometry plays a pivotal role in calculating the area of regular polygons. Specifically, the cotangent function \( \cot\left(\frac{\pi}{n}\right) \) is used in the area formula.
This trigonometric function is crucial because it helps measure the angle between adjacent sides of a polygon. The angle, determined by the number of sides, influences how the sides align and converge, impacting the overall shape and area.
Trigonometry links geometry and angles for precise calculations. As every angle in a polygon varies, the cumulative effect contributes to the polygon's unique shape and thus its area.

• \( \cot\) relates to opposite and adjacent side lengths in triangles formed within the polygon.
• This relation settles the heights and angles, grounding the area calculation reliably and precisely.

Understanding this mathematical insight helps grasp why area adjustments, like side doubling, don't linearly affect the area outcome.

Effects of Doubling Side Lengths in Regular Polygons

When examining the effect of doubling the side length of a regular polygon, it’s essential to explore how geometric properties influence area changes. If you double the length of each side from \( s \) to \( 2s \), the new calculation reveals significant changes:

• The side length input in the area formula is squared \( (s^2) \). Therefore, doubling the side length quadruples this value.
• Doubling causes changes by a factor of \( 4 \), not just \( 2 \), due to the squaring process.
This geometric consideration shows that increases in length have an amplified effect on area. So, if the original area was \( A \), doubling the side creates a new area, \( 4A \), not double \( 2A \).
This demonstrates how intertwined geometric properties affect outcomes beyond basic doubling, illustrating why intuitive guesses may often mislead without considering all factors at play.