Question

Find the area of the shaded region in each figure.Round your answer to 2 decimal places if required.

Answer 1

Divide each region into already known polygons and circles. Then, calculate the area of each figure involved and make the necessary operations to find the area of each shaded region.

1)

Notice that the figure can be divided in a semicircle and a trapeze:

The area of a circle is given by:

`A=\pi\cdot r^2`

And the area of a trapeze is given by:

`A=(1)/(2)(B+b)\cdot h`

Add half the area of a circle of radius 10in plus the area of the trapeze.

2)

The figure can be interpreted as a parallelogram with a missing triangular section:

The base of both the parallelogram and the triangle is 19ft, while the height of the parallelogram is 15ft and the height of the triangle is 7ft.

The area of the parallelogram is given by:

`A=b\cdot h`

The area of the triangle is given by:

`A=(1)/(2)b\cdot h`

(notice that they don't have the same height)

Substract the area of the triangle from the area of the parallelogram to find the area of the shaded region.

3)

This figure is a circle with diameter 12yd with a missing triangular section with its base matching the diameter of the circle.

Since the radius is half the diameter, the radius of the circle is 6yd. On the other hand, one vertex of the triangle lies on the circumference of the circle, then, the distance from that vertex to the center equals the radius of the circle, which is 6yd. Furthermore, the segment from the center of the circle to that vertex of the triangle seems to be perpendicular to the diameter. Then, that segment is also the height of that triangle:

Substract the area of the triangle from the area of the circle to find the area of the shaded region.

4)

The figure is a composition of a square and a parallelogram:

The height of the parallelogram must be 7in, since the height of the whole figure is 18in and 11in correspond to the height of the square.

Compute the area of each figure and then add them. In both cases, the area equals base times height:

`A=b\cdot h`

(in the case of a square, both height and base have the same value).

Area of a trapezoid:

A trapezoid is a quadrilateral where there are two parallel sides, which we call the bases of the trapezoid. The distance between the bases is called the height. Given that the bases have lengths b and B and the height has a length of h, the area is given by:

`A=(1)/(2)(b+B)\cdot h`