1. All you need for this one is the length of each side and the semiperimeter (Half of the perimeter). The formula for the semiperimeter is:
Where a, b, and c are the sides of the triangle, and since they are equilateral triangles, they all have the same side length, which I will call . Our new formula is:
We know that l has to be 8, because it's a three-sided shape with equal side lengths, so our semiperimeter is:
Now, we need to know what's called the inradius, which we use the following equation to get:
Now, we know that s is 12 and a b and c are all 8, so we just plug and chug:
Now, the area of a circle is:
We know the radius, all we have to do is plug and solve:
And there's that one.
2. Notice that the median of the trapezoid fulfills the diameter of circle completely. and the formula for that is:
This is the diameter of the circle, so now we also have the height of the trapezoid and can get the area of the circle. First, we'll get the area of the trapezoid:
Now, we need the area of the Circle:
Now, when you draw the median, you cut the trapezoid into two separate trapezoids, we are interested in the bottom half, where we know the following:
Now we need to subtract the area of the half-circle from the area of the new trapezoid, and we will be left with the areas of the two identical corners:
Now, we divide this new area by 2 to get the area of a single corner:
There you go.