Answer:
Step-by-step explanation:
To find the area of the shaded region, we need to find the area of the hexagon and then subtract the area of the six congruent triangles formed outside the hexagon.
First, let's find the area of the hexagon. A regular hexagon can be divided into six equilateral triangles. The formula for the area of an equilateral triangle with side length \( s \) is:
\[ \text{Area of equilateral triangle} = \frac{\sqrt{3}}{4} \times s^2 \]
Given that the radius of the circle is 4 cm, the side length of the hexagon is also 4 cm (since it's the distance from the center of the circle to a vertex of the hexagon). So, the area of one equilateral triangle is:
\[ \text{Area of one equilateral triangle} = \frac{\sqrt{3}}{4} \times (4)^2 = 4\sqrt{3} \]
Since there are six congruent triangles in the hexagon, the total area of the hexagon is \( 6 \times 4\sqrt{3} = 24\sqrt{3} \) square cm.
Now, let's find the area of one of these triangles. The radius of the circle is also the height of each triangle, and the side length of the hexagon is the base. So, the area of one triangle is:
\[ \text{Area of one triangle} = \frac{1}{2} \times 4 \times 4 = 8 \]
Since there are six congruent triangles in the hexagon, the total area of these triangles is \( 6 \times 8 = 48 \) square cm.
To find the area of the shaded region, we need to find the area of the hexagon and then subtract the area of the six congruent triangles formed outside the hexagon.
First, let's find the area of the hexagon. A regular hexagon can be divided into six equilateral triangles. The formula for the area of an equilateral triangle with side length \( s \) is:
\[ \text{Area of equilateral triangle} = \frac{\sqrt{3}}{4} \times s^2 \]
Given that the radius of the circle is 4 cm, the side length of the hexagon is also 4 cm (since it's the distance from the center of the circle to a vertex of the hexagon). So, the area of one equilateral triangle is:
\[ \text{Area of one equilateral triangle} = \frac{\sqrt{3}}{4} \times (4)^2 = 4\sqrt{3} \]
Since there are six congruent triangles in the hexagon, the total area of the hexagon is \( 6 \times 4\sqrt{3} = 24\sqrt{3} \) square cm.
Now, let's find the area of one of these triangles. The radius of the circle is also the height of each triangle, and the side length of the hexagon is the base. So, the area of one triangle is:
\[ \text{Area of one triangle} = \frac{1}{2} \times 4 \times 4 = 8 \]
Since there are six congruent triangles in the hexagon, the total area of these triangles is \( 6 \times 8 = 48 \) square cm.
