# Show square inscribed in circle has maximum area

1. Sep 6, 2010

### t_n_p

1. The problem statement, all variables and given/known data
Show that the square, when inscribed in a circle, has the largest area of all the 4-sided polygons. Try to show that all sides of a quadrilateral of maximal area have to be of
equal length.

3. The attempt at a solution
How do you start?
I don't get how showing the sides are equal will help prove that the square has maximal area.
Any hints?

2. Sep 6, 2010

### Staff: Mentor

Try to express surface of the inscribed 4-sided polygon as a function of the length of its sides. General case will be more difficult, but for a specific case of rectangle it should be relatively easy to express lengths as a function of one variable.

Once you have that, find maximum.

3. Sep 6, 2010

### t_n_p

I don't quite understand.
Say I have a square, let the side length = a.

area = a^2.
Then maximize by taking derivative?

4. Sep 6, 2010

### rishicomplex

Real easy to prove with coordinate geometry, but you need to know the parametric equation of a circle and the area of a quadrilateral.

Any point on a circle (with origin as centre and radius r) is (r cosx, r sinx)
and area of a quadrilateral with vertices (x1, y1), (x2, y2), (x3, y3), (x4, y4),
is #8 here http://www.mathisfunforum.com/viewtopic.php?id=3301

take four points, put them in the formula, the result becomes obvious

5. Sep 6, 2010

### t_n_p

that gives me the area, but it doesnt prove that it is the maximum area for a 4 sided shape.

6. Sep 6, 2010

### rishicomplex

it does! the result for area of a general quadrilateral in a circle comes to be (assuming radius r and angles A, B, C, D
r2/2[sin(A-B) + sin(B-C) + sin(C-D) + sin(D-A)]
the maximum value of which (2r2) is attained only when each of the angular differences is 90 degrees which results in a square

did u try solving it? you need basic trignometric formulae