Extreme Quadrilateral Areas

Geometry Level 3

We know by Heron's formula that if we know all the side lengths of a triangle, then we can determine the area of the triangle. However, this is not the case for a quadrilateral.

Find the maximum possible area of a quadrilateral with side lengths 3, 4, 5, and 6.

If this area can be expressed as $P \sqrt Q$ for integers $P$ and $Q$ , with $Q$ square-free, find $P+Q.$

The answer is 16.

1 solution

Shreyansh Mukhopadhyay
Jan 11, 2018

By using Bretschneider's Formula

the square of area of the quadrilateral is $\Delta^{2} = (s-a)(s-b)(s-c)(s-d)-abcd\cos^2\left(\frac{B+D}{2}\right)$ .

Therefore the maximum area is when $\cos^2\left(\frac{B+D}{2}\right)$ is minimum.

$\cos^2\left(\frac{B+D}{2}\right)=0$ , when $B+D=180$

So the maximum area is $\Delta =\sqrt{(s-a)(s-b)(s-c)(s-d)}$

This equals to $\sqrt{360}$ , after putting the values given in the question

$P \sqrt{Q} = 6 \sqrt{10}$

So $P+Q=6+10=\boxed{16}$

Why would the area be maximum for a cyclic quadrilateral?

Aman thegreat - 3 years, 4 months ago

I have edited my solution, so now you may understand it better.

Shreyansh Mukhopadhyay - 3 years, 4 months ago

Brahmagupta discovered this formula much before Bretschneider.

Vijay Simha - 3 years, 4 months ago

He found the formula for the area of cyclic quadrilateral, but didn't prove that the maximum area with given side-length is of a cyclic quad.

Shreyansh Mukhopadhyay - 3 years, 4 months ago

Extreme Quadrilateral Areas

The answer is 16.

1 solution

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