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Chapter 5: Problem 33

Find the perimeter of a regular hexagon whose vertices are on the unit circle.

Short Answer

Expert verified
The perimeter of the regular hexagon whose vertices are on the unit circle is 6 units.

Step by step solution

01

Find the angle between the radius and the side of the hexagon

Since the hexagon is regular, we can divide the circle into six equal sectors by drawing radii to each vertex. The central angle of each sector is 360/6 = 60 degrees. The angle between the radius and the side of the hexagon is half of the central angle, so it is 30 degrees.
02

Use trigonometry to find the length of half of the side

Consider the isosceles triangle formed by the radius, the side of the hexagon, and the line segment connecting adjacent vertices. We need to find the length of half of the side. In the isosceles triangle, the angle between the radius and the side of the hexagon is 30 degrees, and since the radius is 1 unit, we have: \(sin(30) = \frac{opposite}{hypotenuse}\) \(sin(30) = \frac{opposite}{1}\) \(opposite = sin(30)\), where "opposite" refers to the length of half of the side of the hexagon.
03

Multiply the length of half of the side by two

Since \(sin(30) = 0.5\), we have: Opposite = 0.5(length of half of the side) = 0.5 To find the length of one side of the hexagon, we multiply the length of half of the side by two: Side length = 0.5 脳 2 = 1
04

Multiply the side length by six

Now that we have the length of one side of the hexagon, we can find the perimeter by multiplying the side length by the number of sides in the hexagon (which is six): Perimeter = side length 脳 6 = 1 脳 6 = 6 units Therefore, the perimeter of the regular hexagon whose vertices are on the unit circle is 6 units.

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