Calculating Area of Shaded Hexagon in a Rectangle
I see a geometric figure with a shaded area that you want to calculate. The overall figure appears to be a rectangle with a smaller, centered hexagon cut out, which is the shaded area. The dimensions given are for the rectangle and the side of the hexagon.
To calculate the area of the shaded hexagon, we need to know the formula for the area of a regular hexagon, which depends on the length of its sides (s). The area (A) of a regular hexagon can be calculated using the following formula:
\[ A = \frac{3\sqrt{3}}{2} \times s^2 \]
Given that the side length (s) of the hexagon is 2 cm, we can substitute this value into the formula to find the area of the hexagon:
\[ A = \frac{3\sqrt{3}}{2} \times 2^2 \]
\[ A = \frac{3\sqrt{3}}{2} \times 4 \]
\[ A = 6\sqrt{3} \text{ cm}^2 \]
Now that we have the area of the hexagon, we can subtract it from the area of the larger rectangle to obtain the area of the shaded region.
The area of the rectangle is found by multiplying its length and width:
\[ \text{Area of rectangle} = \text{length} \times \text{width} \]
\[ \text{Area of rectangle} = 10 \text{ cm} \times 6 \text{ cm} \]
\[ \text{Area of rectangle} = 60 \text{ cm}^2 \]
Now, subtract the area of the hexagon from the area of the rectangle:
\[ \text{Area of shaded region} = \text{Area of rectangle} - \text{Area of hexagon} \]
\[ \text{Area of shaded region} = 60 \text{ cm}^2 - 6\sqrt{3} \text{ cm}^2 \]
This is the final expression for the area of the shaded region. If you need an approximate numerical answer, you can calculate the square root of 3 and multiply by 6, then subtract that result from 60. The square root of 3 is approximately 1.732, so:
\[ \text{Area of shaded region} \approx 60 - 6 \times 1.732 \]
\[ \text{Area of shaded region} \approx 60 - 10.392 \]
\[ \text{Area of shaded region} \approx 49.608 \text{ cm}^2 \]
Therefore, the approximate area of the shaded region is 49.608 square centimeters.
