Question

Need help finding the area of the shaded region. Round to the nearest tenth. Need help ASAP

Answer 1

294.36

Explanation:

Find the area of the entire circle.

Subtract out the area of a sector with 130 degrees for the central angle.

Add the area of the isosceles triangle with an apex angle of 130 degrees.

Area of The entire circle

Area = pi * r^2

Area = 3.14 * 11.1^2

Area = 386.88 m^2

Area of the sector with 130 degrees for a central angle

Area_130 = (130/360) * pi * r^2

Area_130 = (130/360) * 3.14* 11.1^2

Area_130 = 139.71

Area of the triangle

1/2 central angle = 130/2 = 65

Bisect the apex angle so that each half = 65 degrees.

Sin(65) = opposite / hypotenuse

Sin(65) = Opposite / 11.1

11.1 * sin(65) = opposite

opposite = 10.06

This is 1/2 the base so the base = 2*10.06 = 20.12

The height of the triangle is found by cos(65) = adjacent/hypotenuse

hypotenuse = 11.1

Cos(65) = adjacent / hypotenuse

adjacent = hypotenuse * cos(65)

adjacent = 4.69 This is the height of the triangle.

Area of the triangle = 1/2 * 20.12 * 4.69

Area of the triangle = 47.19 m^2

Area of the Shaded Area

Area of entire circle - area of sector + area of triangle

=386.88 - 139.71 + 47.19

=294.36

Note

The area of the triangle could be done using Area = 1/2 * 11.1^2 * (2*sin(65)*cos(65) = 1/2 * 11.1^2 * sin(130) = 47.2 but you may not know all the math to do the area this way.

Answer 2

Answer: 294.4 m²

Explanation:

Separate the shaded region into two parts:

1. The section containing the central angle of 230° (360° - 130°)
2. The triangle with sides 11.1, 11.1 & 20.12 (use Law of Cosines)

`1.\ Area(A)=\pi\ r^2\ \bigg((\theta)/(360)\bigg)\\\\\\.\qquad \qquad =\pi(11.1)^2\bigg((230)/(360)\bigg)\\\\\\.\qquad \qquad =247.3`

`2.\ \text{Use Law of cosines to find the length of the third side.}\\\text{ Then use Heron's formula to find the Area of the triangle.}\\\\s=(11.1+11.1+20.12)/(2)=21.16\\\\\\A=√(s(s-a)(s-b)(s-c))\\\\.\ =√(21.16(21.16-11.1)(21.16-11.1)(21.16-20.12))\\\\.\ =√(2227)\\\\.\ =47.1`

Area of shaded region = Area of (1) + Area of (2)

= 247.3 + 47.1

= 294.4