Question

The polygons are regular polygons. Find the area of the shaded region.​

Answer 1

Answer:
`194.8557 in^2`

The side of a regular hexagon is congruent with the radius of the circle the hexagon is inscribed with. That makes the two triangles in red and green (lines are a bit offset for clarity) equilateral. At this point it's easy to compute the area of an equilateral triangle known the side length l, which is
`A = \frac12 l (l\frac{\sqrt3}2) = \frac{\sqrt3}4 l^2` At this point, the area of the shaded region of one triangle making up the hexagon is obtained by difference (area of the red triangle, minus area of the white:

`A=A_r-A_w = \frac{\sqrt3}4 10^2 -\frac{\sqrt3}45^2 = \frac{\sqrt3}4(10^2-5^2)=\frac{\sqrt3}4* 75\approx 32.47595`

The area of the whole shaded region, is 6 times that, or
`194.8557`

With tasselation:

Join the midpoint of the side of the external hexagon with the end points of the corresponding side of the inner hexagon. You have now created four equilateral squares which are congruent. (yes, my drawing skills are terrible)

The shaded region is made of 18 triangular tiles, each having an area of (see above formula)
`\frac{\sqrt3}4* 5^2 = 18.8253`, for a total of, again, 194.8557 square inch.

Answer 2

The area of the shaded region is 194.8557 square feet.

Identify the shapes and their properties

The two polygons in the image are regular hexagons, which means they have six equal sides and six equal angles.

The side length of each hexagon is given as 5 feet.

Calculate the area of one hexagon

The area of a regular hexagon can be calculated using the formula: Area = (3√3 / 4) * side^2

Plugging in the side length of 5 feet, we get the area of one hexagon: Area = (3√3 / 4) * 5^2 ≈ 64.9519 sqft

Divide the shaded region into smaller shapes

The shaded region can be divided into six isosceles triangles and one central hexagon.

Calculate the area of one triangle

The triangles are isosceles with a base of 5 feet (the side of the hexagon) and a height equal to the apothem of the hexagon.

The apothem of a regular hexagon is the distance from the center of the hexagon to the midpoint of a side. It can be calculated using the formula: apothem = side length * √3 / 2

In this case, the apothem is 5 * √3 / 2 ≈ 4.3301 feet.

The area of each triangle is then: Area = (1/2) * base * height = (1/2) * 5 * 4.3301 ≈ 11.5476 sqft

Calculate the area of the shaded region

The area of the shaded region is equal to the area of six triangles plus the area of the central hexagon minus the area of the small hexagon inside the shaded region.

Area of shaded region = 6 * area of triangle + area of large hexagon - area of small hexagon

Area of shaded region = 6 * 11.5476 + 64.9519 - 64.9519

Area of shaded region ≈ 194.8557 sqft