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A regular hexagon is cut out of a regular decagon. The height of the hexagon is
5ft, and the height of the decagon is
10ft. Find the area of the shaded region.

$9$ . A regular hexagon is cut out of a regular decagon. The height of the hexagon is $5 \mathrm{ft}$ , and the height of the decagon is $10 \mathrm{ft}$ . Find the area of the shaded region.

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Area of quadrilaterals and triangles: word problems

Full solution

Q.

9

. A regular hexagon is cut out of a regular decagon. The height of the hexagon is

5 \mathrm{ft}

, and the height of the decagon is

10 \mathrm{ft}

. Find the area of the shaded region.

Calculate Area Decagon: To find the area of the shaded region, we first need to calculate the area of the decagon and then subtract the area of the hexagon from it.
Find Apothem Decagon: The formula for the area of a regular polygon is $(\text{Perimeter} \times \text{Apothem}) / 2$ . However, we are given the height, not the apothem. For a regular decagon, the apothem is approximately $80.9\%$ of the height. So, the apothem of the decagon is $10\text{ft} \times 0.809$ .
Calculate Perimeter Decagon: Calculating the apothem of the decagon: $10\text{ft} \times 0.809 = 8.09\text{ft}$ .
Unable to Proceed: Now, we need to find the perimeter of the decagon. Since we don't have the side length, we can't calculate the perimeter directly. We need the side length to proceed, which we don't have. So, we're stuck here.

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