The area of the largest triangle that can be inscribed in a semicircle of radius r unit is;
This is equal to 2*r (r = the radius)
If the triangle is an isosceles triangle with an angle of 45 degrees at each end, then the height of the triangle is also a radius of the circle.
A = (1/2)*b*h formula for the area of a triangle becomes:
The base of the triangle is equal to 2*r
The height of the triangle is equal to r
A = (1/2)*2*r*r becomes:
A = r^2
Since the area of the triangle is equal to (1/2)*b*h, and the base remains the same while the height of the triangle is less at any other point on the surface of the semi-circle, then the largest area is when the height equals the radius of the triangle.