What i am struggling with is to get the php function to display the "shape_size_fet" (corresponding to the shape) on the select menu when a shape is selected from the HTML form.
Your coming up against a common misconception of PHP. Many programmers think it can dynamically run form the browser. It can not. PHP only runs on a HTTP request, it is a server side script that only runs on the server, and once the server has finished processing a request (file) it sends the results to your browser. No PHP is passed to your browser and to get PHP to run you have to reload or load a page (http request).
You are more accustomed to Javascripts ability to dynamically change what is being displayed. PHP can not change what is displayed without reloading the page.
Now you can get the effect you are looking for but you have to know how to bring both PHP and javascript together. If the amount of data is small you can store it in hidden elements or javascript variables. If there is large amounts of data that must become accessible to the user (without reloading the page) that is when you turn to AJAX (javascript's http request functions).
In your case I would store the data in an HTML element or javascript variables and arrays, then use javascript to detect the onclick event and display the corresponding content.
I hope I haven't frustrated your efforts with this information. You seem knowledgeable of javascript and if you can get over the hump of marrying php and javascript you will find what you can build limitless.
Now for your code first I should note that $REQUEST is a [man]superglobal[/man] that just stores the data that is in $GET, $POST, and $COOKIE. There will be no data in $_REQUEST unless you submit the form.
And if you're trying going to check the value of an item in $REQUEST, this isn't what you want
if($_REQUEST["Pool_Shape"] == $_REQUEST["oval"]){
It would be like this
if($_REQUEST["Pool_Shape"] == 'oval'){
// do something in php here
}
All super global's are very much like an array so "Pool_Shape" is the index (key) and "oval" is the value. All super global's are this way the differences being that you can access super global's in any scope but most you can not change there values using PHP ($SESSION, and $COOKIE are the only ones you can store values using PHP).
Now in your case I would take that "if" out of there entirely just leaving the select, options, and for loop.
<div id="_Shapes" style="display:none">
<select id="Shapes" name="size_In_Feets">
<option seleted>Please Select Pool Size</option>
<?php
$ovalPoolSizes = showOvalPoolSize();
for($index = 0; $index < count($ovalPoolSizes); $index++){
echo '<option value="'.$ovalPoolSizes[$index].'">'.$ovalPoolSizes[$index].'</option>';
}
?>
</select>
</div>
Then just use javascript to show that <div> when the user clicks the corresponding radio button.