Html – Curved corner border for a div

Wow, this problem is popular. It's based on a misunderstanding in the vertical-align property. This excellent article explains it:

Understanding vertical-align, or "How (Not) To Vertically Center Content" by Gavin Kistner.

“How to center in CSS” is a great web tool which helps to find the necessary CSS centering attributes for different situations.

In a nutshell _{^{(and to prevent link rot)}}:

• Inline elements (and only inline elements) can be vertically aligned in their context via vertical-align: middle. However, the “context” isn’t the whole parent container height, it’s the height of the text line they’re in. jsfiddle example
• For block elements, vertical alignment is harder and strongly depends on the specific situation:
  • If the inner element can have a fixed height, you can make its position absolute and specify its height, margin-top and top position. jsfiddle example
  • If the centered element consists of a single line and its parent height is fixed you can simply set the container’s line-height to fill its height. This method is quite versatile in my experience. jsfiddle example
  • … there are more such special cases.

2015 update: the flexbox approach

There are two other answers briefly mentioning flexbox; however, that was more than two years ago, and they don't provide any examples. The specification for flexbox has definitely settled now.

Note: Though CSS Flexible Boxes Layout specification is at the Candidate Recommendation stage, not all browsers have implemented it. WebKit implementation must be prefixed with -webkit-; Internet Explorer implements an old version of the spec, prefixed with -ms-; Opera 12.10 implements the latest version of the spec, unprefixed. See the compatibility table on each property for an up-to-date compatibility status.

(taken from https://developer.mozilla.org/en-US/docs/Web/Guide/CSS/Flexible_boxes)

All major browsers and IE11+ support Flexbox. For IE 10 or older, you can use the FlexieJS shim.

To check current support you can also see here: http://caniuse.com/#feat=flexbox

Working example

With flexbox you can easily switch between any of your rows or columns either having fixed dimensions, content-sized dimensions or remaining-space dimensions. In my example I have set the header to snap to its content (as per the OPs question), I've added a footer to show how to add a fixed-height region and then set the content area to fill up the remaining space.

html,
body {
  height: 100%;
  margin: 0;
}

.box {
  display: flex;
  flex-flow: column;
  height: 100%;
}

.box .row {
  border: 1px dotted grey;
}

.box .row.header {
  flex: 0 1 auto;
  /* The above is shorthand for:
  flex-grow: 0,
  flex-shrink: 1,
  flex-basis: auto
  */
}

.box .row.content {
  flex: 1 1 auto;
}

.box .row.footer {
  flex: 0 1 40px;
}

<!-- Obviously, you could use HTML5 tags like `header`, `footer` and `section` -->

<div class="box">
  <div class="row header">
    <p><b>header</b>
      <br />
      <br />(sized to content)</p>
  </div>
  <div class="row content">
    <p>
      <b>content</b>
      (fills remaining space)
    </p>
  </div>
  <div class="row footer">
    <p><b>footer</b> (fixed height)</p>
  </div>
</div>

In the CSS above, the flex property shorthands the flex-grow, flex-shrink, and flex-basis properties to establish the flexibility of the flex items. Mozilla has a good introduction to the flexible boxes model.