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Chapter 8: Problem 36

Each side of regular hexagon \(A B C D E F\) is 18 feet long. What is the length of the diagonal \(\overline{B D}\) ? Explain your reasoning. CAN'T COPY THE GRAPH

Short Answer

Expert verified
The length of diagonal \(\overline{BD}\) is 36 feet.

Step by step solution

01

Understanding the Problem

We need to find the length of the diagonal \(\overline{BD}\) in a regular hexagon where each side measures 18 feet. In a regular hexagon, all sides and angles are equal.
02

Hexagon Properties

In a regular hexagon, a diagonal connecting two vertices, such as \(\overline{BD}\), is often twice the length of a side. This is because the opposite vertices are separated by two sides of the hexagon.
03

Calculate the Diagonal Length

Since each side of the regular hexagon is 18 feet, the diagonal \(\overline{BD}\) is twice the length of a side. Therefore, \(\overline{BD} = 2 \times 18 = 36\) feet.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hexagon Properties
A hexagon is a polygon with six sides, and a regular hexagon means all sides and angles are the same. In any regular hexagon, you can notice these distinct features:
  • Equal Sides: All six sides hold the same length. This simplifies calculations because the whole hexagon can be divided into congruent parts.
  • Angle Sum: The interior angles sum up to \(720^{\circ}\). Each interior angle measures \(120^{\circ}\) in a regular hexagon.
  • Symmetry: Regular hexagons exhibit rotational symmetry and six lines of symmetry, making shape-identifying patterns clearer.
  • Diagonals: A regular hexagon has three distinct diagonals connecting non-adjacent vertices.
This foundational understanding of hexagons is essential for resolving geometry problems involving these six-sided figures.
Diagonals in Polygons
Diagonals are lines connecting non-consecutive vertices in a polygon. Understanding their formation in any polygon is key:
  • Calculation: For an \(n\)-sided polygon, the total number of diagonals is given by \(\frac{n(n - 3)}{2}\).
  • Example in Hexagons: For a hexagon, \(n = 6\), so it contains \(\frac{6(6 - 3)}{2} = 9\) diagonals.
Examining diagonals helps in calculating distances and understanding polygon structures. In a regular hexagon, diagonals contribute significantly to pattern recognition and make it easier to divide the hexagon into familiar, equal parts. These diagonals can further be cross-referenced with connecting side lengths to solve longer distance calculations.
Regular Polygons
Regular polygons are symmetrical shapes with equal sides and angles, leading to predictable geometric properties:
  • Angles: For any regular polygon, each interior angle can be calculated as \(\frac{(n - 2) \times 180}{n}\) degrees. This uniformity aids in creating balanced compositions.
  • Equal Side Lengths: All side lengths are the same, making perimeter and area computations straightforward.
  • Diagonals and Apothem: Regular polygons can be split into isosceles triangles or other known shapes by their diagonals. The apothem, a line from the center perpendicular to a side, is often used for area calculations.
These traits simplify geometry problems, as they remove irregularities that typically complicate irregular polygon problems. Regular polygons like the hexagon are used often in design and architecture due to their pleasing aesthetics and uniformity.

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