It is easy to skip the sunscreen when it isn't sunny outside but according to Dr. Judith Reichman, "even when the sun doesn't appear to be shining brightly, its rays are there, working their damage." 
There are two types of sun's rays which reach the earth: UVB and UVA. UVB rays cause immediate damage to your skin. These rays are responsible for tanning or burning your skin. UVA rays are those that are responsible for aging your skin and are able to reach your skin through clouds, smog or glass. The American Academy of Dermatologists (AAD) indicates that up to 80 percent of the sun's UV rays can pass through clouds. 
The UV level is an international scale ranging from 1 to 20 and indicates how much UV radiation reaches the earth's surface. A level of 3 or above is considered high enough to damage the skin. Some UV weather forecasts base the UV level on a clear sky, not taking into account clouds, rain or pollution. It gives you the maximum amount of UV rays expected during the day (which would normally occur midday.)
The Effect of Clouds
Clouds can reduce or increase UV levels. According to Vernon Carr, in an interview with ABC Science in Australia, "UV is affected by many things. One of those is cloud, but it depends upon what type of cloud it is, the thickness of the cloud and whether it is deep right through the atmosphere. UV radiation can penetrate through thin cloud, so you can still get high levels of UV at ground level on overcast days." 
Carr also explains that sometimes UV rays can reflect off the cloud's edge, intensifying the level of UV rays. Therefore, partly cloudy days can be misleading; you believe you are protected from the sun's rays when behind the cloud but you may, in fact, be getting stronger UV rays because they are bouncing off the clouds.
How Much Sunscreen Do You Need?
Most people don't use enough sunscreen, according to the AAD, applying only
½ of the recommended amount. An adult should use approximately one ounce of sunscreen to cover exposed areas - one ounce is the size of a shot glass. You may need more or less depending on your body size.
Remember to use sunscreen on all exposed areas of your body, including your face, ears, arms and hands. It should be applied 15 to 30 minutes before going outside. Your lips should be protected with a lip balm with an SPF of at least 30. Sunscreen should be reapplied every 2 hours or after swimming or sweating to ensure the most protection.
These guidelines should be followed year round, no matter what the weather. Protecting your skin will not only help you look younger, it will help protect you from developing skin cancer.
 "Do I Need Sunscreen on a Day with Extreme UV Levels if it's Cloudy?" 2012, Jan 24, Genelle Weute, ABC Science
 "It's a Cloudy Day. Should I Still Use Sunscreen," 2004, May 24, Dr. Judith Reichman, Today.MSNBC.com
 "Sunscreens," Date Unknown, Staff writer, American Academy of Dermatology