Check if a latitude and longitude is within a circle google maps

For such short distances, and when the accuracy doesn't have to be exact to the centimeter, you can treat the surface of the earth as flat. Calculate a conversion from degrees to kilometers at the latitude of the center point, then the Pythagorean theorem can be used to get the distance:

function arePointsNear(checkPoint, centerPoint, km) {
  var ky = 40000 / 360;
  var kx = Math.cos(Math.PI * centerPoint.lat / 180.0) * ky;
  var dx = Math.abs(centerPoint.lng - checkPoint.lng) * kx;
  var dy = Math.abs(centerPoint.lat - checkPoint.lat) * ky;
  return Math.sqrt(dx * dx + dy * dy) <= km;
}

Demo: http://jsfiddle.net/Guffa/57gQa/

Note: The code doesn't take into consideration if you are passing the 0/360 longitude. If that is the case, you would have to normalize the longitudes first.

All you need is a little spherical trig

First you need the central angle theta of the arc subtended by your distance (L = 10 km).

L = theta*r

where r is the radius of the earth (6378.135 km)

Now, if the central angle between the point of interest and your center point is < theta, it is inside your circle. Call this angle theta_p.

Here's a diagram illustrating a spherical triangle: spherical triangle image http://en.wikipedia.org/wiki/File:Spherical_trigonometry_basic_triangle.svg

edit - sorry, apparently I don't know how to link to an image?? Here's the URL: http://en.wikipedia.org/wiki/File:Spherical_trigonometry_basic_triangle.svg

In this case, two of the sides of the spherical triangle (call them a, b) are the difference in longitude and difference in latitude of the points respectively. The included angle C is 90 degrees (angle between lines of longitude and lines of latitude.

The spherical trig law of cosines is:

cos(c) = cos(a)*cos(b) + sin(a)*sin(b)*cos(C)

c is the central angle between your points, which we earlier called theta_p

edit - this solution isn't limited to small distance WRT the radius of the earth, as the other suggestions are.