You are to build a fence using two straight pieces and a piece that is an arc of a circle. The straight piece

You are to build a fence using two straight pieces and a piece that is an arc
of a circle. The straight pieces should have equal length, and the centre of
the circle should be the point where the two straight pieces meet, but you
get to choose the angle between those straight pieces. The total amount
of fencing you can use is 10 meters. What dimensions will maximize the
area enclosed?

Area of sector is A=1/2*r^2*phi

Circumference of sector U=2*r+r*phi=10

Resolve for phi, phi=(10-2*r)/r
and insert into area A

A=1/2*r^2*(10-2*r)/r=1/2*r(10-2*r)=5r-r…

Find Maximum:
dA/dr=0, 5-2*r=0 -> r=2.5 -> phi=2, A=6.25