To find the area, but the easiest approach is using Brahmagupta’ s formula1:

If a quadrilateral of side lengths a, b, c, d can be inscribed in a circle, then its area is given by
A = sqrt<(s − a)(s − b)(s − c)(s − d)>, where s =
(a + b + c + d)/2

In this case, a = 2*radius
You know b and d are equal, and you can write a relationship, I think between b and c (c= d - 2b*cosAngle) and you can figure the cosine of the angle in terms of the secants.

Then maximize the area, set to zero, and solve for b or c.

Could be some nasty math. I don't see an easier analytic method.

find the dimension of the trapezoid of greatest area that can be inscribed in a semicircle of radius r

To find the dimensions of the trapezoid of greatest area that can be inscribed in a semicircle of radius r, we can follow these steps:

1. Start by visualizing the problem. Draw a semicircle with radius r. Then draw a trapezoid inside the semicircle such that its bases are parallel to the diameter of the semicircle.

2. Let the length of the parallel bases of the trapezoid be a and b.

3. By symmetry, we can assume that the longer base (b) is positioned horizontally at the bottom and the shorter base (a) is positioned vertically at the top.

4. Use Pythagoras' theorem to find the height (h) of the trapezoid. Consider the right-angled triangle formed by the radius, the height, and one of the bases. The hypotenuse is the radius (r), and one side is the height (h). Using Pythagoras' theorem, we have: r^2 = (a/2)^2 + h^2.

5. Rearrange the formula derived from Pythagoras' theorem to express h in terms of a: h = sqrt(r^2 - (a/2)^2).

6. Determine the length of the other diagonal (c) of the trapezoid. Since the trapezoid is inscribed in the semicircle, both diagonals will be equal to the diameter, which is 2r.

7. Use the formula for the area of a trapezoid to find the area (A) in terms of a, b, and h. The area of a trapezoid is given by A = (a + b)/2 * h.

8. Substitute the expression for h derived in step 5 into the area formula from step 7. This will give the area of the trapezoid solely in terms of a and b.

9. To maximize the area, differentiate the area formula with respect to either a or b, set the derivative equal to zero, and solve for one of the variables. This will yield the critical values for a or b.

10. Calculate the corresponding value for the other variable using the relationship between a and b.

11. Evaluate the area using the values obtained for a and b. This will give you the maximum area of the trapezoid that can be inscribed in the semicircle.

Remember, this process may involve some complex math, so it might be helpful to use a computer algebra system or calculator to handle the calculations.

It's also worth noting that the solution obtained using this method will be an approximation, as finding the exact dimensions of the trapezoid of maximum area may involve non-elementary functions.