# How much is the max area of a trapezoid if its perimeter is known?

If we have already known the perimeter of a trapezoid, what is its maximum area?

First, the equation I used to calculate the area of a trapezoid is: $$A = \frac{x+y}{2} \times h$$

For my question, I suppose that the perimeter is $C$ and I have the relationship between the perimeter and bases and legs: $$C = x + y + a + b$$ In this equation, $x$ and $y$ are the lengths of the bases and $a$ and $b$ are the lengths of the legs. Then we have these relationships:

$$h = a \times sin{\alpha} = b \times sin\beta$$ $$y + a\times cos\alpha + b \times cos\beta = x$$  wherein $\alpha$ is the angle between base $x$ and leg $a$ and $\beta$ is the angle between base $y$ and leg $b$. $h$ is the length of the height. Then I do not know how to continue my work.

Further thinking: If the sum of lengths of one base and two legs are fixed, that is:

$$C = x + a + b$$

what is the maximum area of the trapezoid? Anticipating your reply.

-
This sounds like a homework problem, and you didn't provide any work. Further, the question is unclear. Are you speaking merely of the optimization of the area of a rectangle (in which case, why the ladder?), or is the ladder leaning up against something? – Jebruho Dec 16 '12 at 6:30
@Jebruho I have already solved the problem of rectangle. Then I want to extend the problem to the problem of a ladder. I am trying to solve this but failed. I have a feeling that there must be a maximum value of the area. – Chuck Wang Dec 16 '12 at 6:35
@Jebruho I have edited my question. – Chuck Wang Dec 16 '12 at 6:50
It might be a language thing, but "expecting your reply" might be considered by some as rude. Perhaps you meant "anticipating" instead of "expecting." "Expectation" means you think people ought to reply, "anticipation" would mean that you look forward to replies. – Thomas Andrews Dec 16 '12 at 8:21
@Jebruho Anticipating your reply. – Chuck Wang Dec 16 '12 at 8:32

## 2 Answers

Hint: for any trapezoid, show there is a rectangle of equal area and smaller perimeter. Therefore, by stretching, there is a rectangle of equal perimeter but greater area.

-
Thanks for your answer. – Chuck Wang Dec 16 '12 at 8:06
Hey Mr. Andrews, I have modified my question and added some more information. I need your help. – Chuck Wang Dec 16 '12 at 8:12

Imagine rotating the legs about their midpoints to make them perpendicular to the bases. That doesn't change the area of the trapezoid, but it decreases the length of the legs. Now move the bases apart, extending the legs until the sum of their lengths is as before the rotation. Now you've increased the area without changing the perimeter. It follows that the optimal trapezoid is a rectangle.

-
Thanks for you reply. I have modified my question and added another question. Can you help me? – Chuck Wang Dec 16 '12 at 8:13