Question

Among all shapes with the same perimeter a circle has the largest area . Prove it

Answer 1

The question can be asked such as "Find the plane curve which encloses given area with shortest perimeter". Then by parametric formulation of coordinates (),()
x
(
t
)
,
y
(
t
)
and assuming non-selfintersecting curve the area can be written as
=12∫21(′−′)
A
=
1
2
∫
t
1
t
2
(
x
y
′
−
x
′
y
)
d
t
and the perimeter
=∫21′2+′2‾‾‾‾‾‾‾‾√
I
=
∫
t
1
t
2
x
′
2
+
y
′
2
d
t
Then the Euler-Lagrange becomes
−(′′2+′2‾‾‾‾‾‾‾‾√)−2(′+())=0
−
d
d
t
(
x
′
x
′
2
+
y
′
2
)
−
λ
2
(
y
′
+
d
d
t
(
y
)
)
=
0
−(′′2+′2‾‾‾‾‾‾‾‾√)−2(−′−())=0
−
d
d
t
(
y
′
x
′
2
+
y
′
2
)
−
λ
2
(
−
x
′
−
d
d
t
(
x
)
)
=
0
If
d
s
is the element of arc the equations can be reduced to
22+=0
d
2
x
d
s
2
+
λ
d
y
d
s
=
0
22−=0
d
2
y
d
s
2
−
λ
d
x
d
s
=
0
which has the solution
()=1+1cos()+2sin()
x
(
t
)
=
K
1
+
C
1
cos
⁡
(
λ
s
)
+
C
2
sin
⁡
(
λ
s
)
()=2−2cos()+2sin()
y
(
t
)
=
K
2
−
C
2
cos
⁡
(
λ
s
)
+
C
2
sin
⁡
(
λ
s
)
It can be further simplified by "Angle sum and difference identities"
()=1+sin(+)
x
(
t
)
=
K
1
+
D
sin
⁡
(
λ
s
+
α
)
()=2−cos(+)
y
(
t
)
=
K
2
−
D
cos
⁡
(
λ
s
+
α
)
where sin()=1
D
sin
⁡
(
α
)
=
C
1
and cos()=2
D
cos
⁡
(
α
)
=
C
2
; and this solution represents a circle of radius
D
and centre (1,2)