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Chapter 29: Problem 557

Find the area of a regular hexagon if one side has the length of 6

Short Answer

Expert verified
The area of the regular hexagon is \(54\sqrt{3}\).

Step by step solution

01

Find the area of one equilateral triangle

To find the area of one equilateral triangle, we can use the formula: Area = (side length^2 * sqrt(3)) / 4 In this exercise, the side length is 6. Area = (6^2 * sqrt(3)) / 4 Area = (36 * sqrt(3)) / 4 Area = 9 * sqrt(3)
02

Find the area of the hexagon

Now that we've found the area of one equilateral triangle, we can find the area of the hexagon by multiplying the area of one triangle by 6: Hexagon Area = 6 * (9 * sqrt(3)) Hexagon Area = 54 * sqrt(3) The area of the regular hexagon is 54 * sqrt(3).

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Key Concepts

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

Regular Hexagon
A regular hexagon is a six-sided polygon where each side has the same length, and all the interior angles are equal. This symmetry makes regular hexagons an interesting topic in geometry. In nature, we see regular hexagons in the honeycomb patterns of beehives. In mathematics, their symmetry and repeating structure make calculations involving their properties particularly straightforward.

Regular hexagons are composed of six equilateral triangles. By dividing a hexagon into these triangles, calculations become simpler. Understanding the regular hexagon is crucial as it provides a foundation for advanced geometric concepts.
Equilateral Triangle
An equilateral triangle is a triangle where all three sides are of equal length and all three angles are equal, each measuring 60 degrees. This is not just a basic geometric shape but also the building block of regular hexagons.

Each equilateral triangle has important properties making it predictable and easy to work with in calculations, especially when divided into smaller segments. In the context of a regular hexagon, every side of the hexagon is also a side of these six equilateral triangles. Mastering equilateral triangles unlocks more complex polygon understanding.
Area Calculation
The area of a shape is a measure of the space it occupies. In geometry, calculating the area of a shape like a hexagon involves breaking it down into smaller, manageable parts鈥攊n this case, equilateral triangles. For an equilateral triangle, the area can be calculated using the formula:
  • Area = \( \frac{{\text{{side length}}^2 \times \sqrt{3}}}{4} \)
This formula involves the side length and an essential constant, \(\sqrt{3}\), reflecting the triangle's angles and proportions.

Once the area of one equilateral triangle is established, multiply it by six to find the total area of the hexagon. By understanding this step-by-step method, you'll improve your skills in tackling both simple and complex geometric problems.
Side Length
The side length is a fundamental measurement in geometry, particularly for polygons like the hexagon and triangle. It refers to the length of one side of a polygon. In a regular hexagon, the side length is consistent across all sides, facilitating calculations.

In the case of the equilateral triangle, the side length directly influences the area. For example, if the side length is 6, you calculate the triangle's area as:
  • First, square the side length (\(6^2\)) to get 36
  • Then, multiply by \(\sqrt{3}\), and finally divide by 4 to arrive at \(9 \times \sqrt{3}\)
Having a clear grasp of the side length鈥檚 role helps in both drawing and solving geometric shapes accurately.

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Most popular questions from this chapter

The side of a regular pentagon is 20 inches in length, (a) Find, to the nearest tenth of an inch, the length of the apothem of the pentagon. (b) Using the result obtained in part (a), find, to the nearest ten square inches, the area of the pentagon.

Compute \(\tan \pi / \mathrm{n}\) and compare with \(\pi / \mathrm{n}\) for a) \(\mathrm{n}=9\); b) \(n=12\); c) \(\mathrm{n}=18\) d) \(\mathrm{n}=36\); e) \(n=180\)

Find the area of a regular hexagon inscribed in a circle of radius \(\mathrm{r}\). Calculate the area explicitly when a) \(\mathrm{r}=4\), b) \(\mathrm{r}=9\), c) \(\mathrm{r}=16\) d) \(\mathrm{r}=25\).

Prove that the area bounded by a regular polygon of \(n\) sides circumscribed about a circle with a radius of length \(\mathrm{r}\) is given by the formula \(\mathrm{A}=\mathrm{nr}^{2} \tan \pi / \mathrm{n}\).

Prove that the area bounded by a regular polygon of \(\mathrm{n}\) sides which is inscribed in a circle with radius of length \(\mathrm{r}\) is given by the formula \(\mathrm{A}=\left(\mathrm{nr}^{2}\right) / 2 \sin 2 \pi / \mathrm{n}\)
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