The largest triangle in a semicircle has his base equal to the diameter of the semicircle
⇒base of Δ =diameter of semicircle
                    = 2r units
∴Base of Δ = 2r
The triangles height must be the radius of the semicircle
⇒ height of the Δ = radius of the semicircle
                            = r units
Now we have
base = 2r
height = r
area of Δ = (1/2) b×h
                = (1/2) 2r × r  units² 
                = 2r²/2
                = r²
∴Area of the largest triangle that can be inscribed in the given semicircle is r² units²