If you want to use an entire DIV to toggle a radio, you need to use some JS. If you just want to make the radio selectable via the text next to it and the area between the radio and the text label, use this:
<p><label><input type="radio">Click me.</label></p>
You do not need quotes around the type, but it's a good idea to use them, anyway. I prefer to use the 'p' tags around my inputs so that I can better automate error correction. It isn't necessary, and you'll hear a lot of people say, "That p tag isn't necessary."
You will also want to open and close with the 'fieldset' tag and perhaps a 'legend' tag. You can see specifications at w3c 4.10, on HTML5 forms.
You may also want to use the 'value' tag. For example:
<form method="post">
enctype="application/x-www-form-urlencoded"
action="https://coke.example.com/order.cgi">
<p><label>Order number: <input name="orderID"></label></p>
<fieldset>
<legend>Drink</legend>
<p><label><input type="radio" value="coke">Do you want a Coke?</label></p>
<p><label><input type="radio" value="pepsi">Or do you want a Pepsi?</label></p>
</legend>
</fieldset>
</form>
With the code I have submitted, I was unable to reproduce your problem, sir.