A regular hexagon is circumscribed about a circle of radius 5. If the perimeter of the hexagon is 38, what is the area of the hexagon? O 75 sq. units O 85 sq. units O 65 sq. units O 95 sq. units
The Correct Answer and Explanation is:
Let’s break down the problem and calculate the area of the hexagon step by step.
Step 1: Understanding the relationship between the hexagon and the circle
A regular hexagon can be circumscribed about a circle, meaning the circle touches the hexagon at each of its six sides. The radius of the circle is the distance from the center of the hexagon to the middle of each side.
Given:
- The radius of the circle, r=5r = 5r=5 units.
- The perimeter of the hexagon is 38 units.
Step 2: Find the side length of the hexagon
The perimeter of a regular hexagon is the sum of the lengths of all its sides. Since the hexagon has six equal sides, we can divide the perimeter by 6 to find the length of one side: Side length of the hexagon=Perimeter of hexagon6=386=6.33 units.\text{Side length of the hexagon} = \frac{\text{Perimeter of hexagon}}{6} = \frac{38}{6} = 6.33 \text{ units}.Side length of the hexagon=6Perimeter of hexagon=638=6.33 units.
Step 3: Relationship between side length and radius
In a regular hexagon, the radius of the circumscribed circle is the same as the length of each side of the hexagon. Therefore, the side length of the hexagon is equal to the radius of the circle: Side length of hexagon=5 units.\text{Side length of hexagon} = 5 \text{ units}.Side length of hexagon=5 units.
It seems there is a mismatch with the perimeter provided. The perimeter calculated earlier suggests a side length of 6.33 units, but the actual side length from the circle’s radius should be 5 units. This indicates that the perimeter of 38 units is for an incorrect hexagon configuration or the problem is misstated.
Step 4: Area of the regular hexagon
The area of a regular hexagon can be found using the formula: A=332×s2,A = \frac{3\sqrt{3}}{2} \times s^2,A=233×s2,
where sss is the side length of the hexagon. Substituting s=5s = 5s=5: A=332×52=332×25=7532.A = \frac{3\sqrt{3}}{2} \times 5^2 = \frac{3\sqrt{3}}{2} \times 25 = \frac{75\sqrt{3}}{2}.A=233×52=233×25=2753.
Approximating 3≈1.732\sqrt{3} \approx 1.7323≈1.732: A≈75×1.7322=129.92≈64.95 square units.A \approx \frac{75 \times 1.732}{2} = \frac{129.9}{2} \approx 64.95 \text{ square units}.A≈275×1.732=2129.9≈64.95 square units.
Conclusion:
The area of the hexagon is approximately 65 square units. Therefore, the correct answer is: 65 sq. units.\boxed{65 \text{ sq. units}}.65 sq. units.
