Problem 45 - Basic Technical Mathematics with Calculus

Question

In Exercises \(41-46,\) solve the given problems. In Exercises \(41-44\) refer to Fig. 4.42. In Exercises 41-43, express the other parts in terms of the listed parts, which are assumed known. Find the exact area of a circle inscribed in a regular hexagon (the circle is tangent to each of the six sides) of perimeter 72

Step-by-Step Solution

Answer

The exact area of the inscribed circle is \( 108\pi \).

Step 1: Understand the Regular Hexagon

A regular hexagon can be divided into 6 equilateral triangles. Each side of the hexagon is equal to the radius of the inscribed circle.

Step 2: Calculate Side Length

Since the perimeter of the hexagon is 72 and there are six equal sides, each side of the hexagon is \[ \frac{72}{6} = 12. \]

Step 3: Relate to Inscribed Circle

The radius of the inscribed circle in a regular hexagon is equal to the height of one of its equilateral triangles, and is \( \frac{\sqrt{3}}{2} \) times the side length of the triangle (or hexagon). Hence, the radius \( r = \frac{\sqrt{3}}{2} \times 12 = 6\sqrt{3} \).

Step 4: Calculate the Area of the Circle

The area of a circle is given by the formula \( A = \pi r^2 \). Substitute the expression for \( r \) to find the area: \[ A = \pi (6\sqrt{3})^2 = 108\pi. \]

Step 5: Conclude with the Exact Area

Thus, the exact area of the circle is \( 108\pi \).

Key Concepts

Regular Hexagon

A regular hexagon is a unique and interesting shape. It consists of six equal-length sides, and each interior angle is 120 degrees. This symmetry makes a regular hexagon easy to work with in geometry. When you break a regular hexagon down, you get six equilateral triangles.

• The sides are equal, giving it a neat sense of order and symmetry.
• Each set of opposite sides is parallel.

Understanding these properties helps us solve problems involving hexagons by simplifying them into more familiar shapes like triangles.
This is especially useful when looking for area, perimeter, or other properties.

Inscribed Circle

An inscribed circle inside a regular hexagon touches each of the hexagon's sides precisely once. This circle is also known as the incircle.

• The radius of this circle is a vital link to understanding the hexagon itself.
• In a regular hexagon, the radius is equal to the height of the equilateral triangles that make up the hexagon.

If you imagine a circle perfectly fitting inside this shape, you realize that each side of the hexagon is tangent to the circle. This means the radius can be a great tool in understanding distances and dimensions.

Perimeter

In geometry, the perimeter is the total distance around a two-dimensional shape, like our hexagon. For a regular hexagon, where all sides are equal, finding the perimeter is straightforward: multiply the length of one side by six.

• If the total perimeter is known, like in this problem, you can easily find the length of one side by dividing the total by six.
• In the given problem, with a perimeter of 72, each side of the hexagon is 12 units.

This simplicity is a key reason why regular hexagons are often used in problems involving perimeter.

Equilateral Triangles

Equilateral triangles are an integral part of geometry due to their perfect symmetry, with all sides and angles equal. In a regular hexagon, each part of the hexagon can be seen as one of these triangles.

• The three sides of each triangle are the same, which is a property that helps in calculations like finding height or radius.
• These triangles connect directly to features like the inscribed circle.

Knowing that a regular hexagon divides neatly into six equilateral triangles simplifies the process of finding dimensions, like the radius of an inscribed circle, presenting an essential step in solving our original problem.