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how to find the perimeter of this compound shape?

The perimeter is the the length of the path all around the figure.
Since it is a length, it has units of length (such as miles, kilometers, or feet), and not units of area (such as square miles, square kilometers, or square feet).

The figure is made up of two shapes.
A triangle at the bottom.
A semicircle at the top.
The dashed line is not part of the path around the figure, so its length is not part of the perimeter. The perimeter is the length of the path drawn as a solid line.

The left side of the triangle has length 10.2 mi.
The sloping side of the triangle has length 18.7 mi.

We need to calculate the length of the semicircle.
That is the same as half of the circumference of a circle.

The circumference of a circle is pi * d, where d = diameter of the circle.
Since we want the circumference of the semicircle, we want half of the circumference of the circle, so we need (1/2) * pi * d.
We can use 3.14 for pi.
d, the diameter, is 15.6 mi.

(1/2) * pi * d = (1/2) * 3.14 * 15.6 mi = 24.5 mi

The half-circle has length 24.5 mi.

Now we can add all the components to find the perimeter of the figure.

P = 10.2 mi + 18.7 mi + 24.5 mi

P = 53.4 mi

The perimeter of the figure is 53.4 mi.