A regular hexagon is inscribed in a circle with radius 10 cm. We need to find: 1. The length of one side of the hexagon.

1. Problem Description

A regular hexagon is inscribed in a circle with radius 10 cm. We need to find:

1. The length of one side of the hexagon.

2. The perimeter of the hexagon.

3. The area of the hexagon.

4. The area of the circle.

5. The percentage of the circle's area occupied by the hexagon. We should use $\pi \approx 3.14$ and round answers to 2 decimal places.

2. Solution Steps

1. Length of one side of the hexagon:

A regular hexagon inscribed in a circle has sides equal to the radius of the circle. Therefore, the length of one side of the hexagon is 10 cm.

2. Perimeter of the hexagon:

The perimeter of a regular hexagon is 6 times the length of one side.

$Perimeter = 6 \times side$

$Perimeter = 6 \times 10$

$Perimeter = 60$ cm.

3. Area of the hexagon:

A regular hexagon can be divided into 6 equilateral triangles. The area of an equilateral triangle with side $s$ is given by:

$Area_{triangle} = \frac{\sqrt{3}}{4} s^2$

Since the side of each equilateral triangle is 10 cm,

$Area_{triangle} = \frac{\sqrt{3}}{4} (10)^2 = \frac{\sqrt{3}}{4} \times 100 = 25\sqrt{3}$

The area of the hexagon is 6 times the area of one equilateral triangle.

$Area_{hexagon} = 6 \times 25\sqrt{3} = 150\sqrt{3}$

Since $\sqrt{3} \approx 1.732$,

$Area_{hexagon} \approx 150 \times 1.732 = 259.8$ cm$^2$.

4. Area of the circle:

The area of a circle with radius $r$ is given by:

$Area_{circle} = \pi r^2$

$Area_{circle} = 3.14 \times (10)^2 = 3.14 \times 100 = 314$ cm$^2$.

5. Percentage of the circle's area occupied by the hexagon:

$Percentage = \frac{Area_{hexagon}}{Area_{circle}} \times 100$

$Percentage = \frac{259.8}{314} \times 100$

$Percentage \approx 0.8274 \times 100 = 82.74 \%$

Rounding to two decimal places, the percentage is 82.74%.

3. Final Answer

1. Length of one side of the hexagon: 10 cm

2. Perimeter of the hexagon: 60 cm

3. Area of the hexagon: 259.80 cm$^2$

4. Area of the circle: 314.00 cm$^2$

5. Percentage of the circle's area occupied by the hexagon: 82.74%