How To Find The Surface Area Of A Cylinder
Cylinder definition
A cylinder has two faces, two curved edges where the curved wall meets the end circles, and a curved surface stretching between the two circular ends.
You encounter cylinders in everyday activities, like having a can of soda, opening a metal can of food, or smacking your friends with a cylindrical pool noodle. A cylinder mathematically is a three dimensional object, a pair of congruent circles separated by a curved surface.
A cylinder is a three-dimensional solid, having height (h), width (w) or diameter (D) and length (l). Since it is three-dimensional, it has surface area instead of simply area (area is generally associated only with two-dimensional shapes, like a circle or rectangle).
Cylinders typically have perpendicular sides to their ends, making them right cylinders. Cylinders can be oblique, too. Their two circular ends may not line up, so the wall or curved surface is leaning, like the oblique cylinder of the famous Leaning Tower of Pisa.
Surface area of a cylinder
When we talk about the surface area of a cylinder, there are really two surface areas we are talking about: the lateral surface area and the total surface area. Total surface area is commonly referred do as the surface area.
If you are asked to find the surface area of a cylinder, then you want to find the areas of the two ends and the curved surface.
Make certain you understand the connection between radius, diameter, and ππ, since they all play a role in determining the surface area of the right cylinder.
The diameter and radius of the cylinder emerge from the two circles that are usually considered the bases, or top and bottom of the cylinder, though no mathematical reason exists for the cylinder to stand up. Think of tank cars in a train; they are cylinders "on their sides," their bases at either end.
All you are doing in calculating surface area is measuring the area of the two circles, the height, h, of the cylinder, and using π to relate them.
Surface area of a cylinder formula
The formula for the surface area of a cylinder is:
In this formula, a, is the total surface area, r is the radius of the circles at both ends, h is the height, and π is the irrational number that we simplify and shorten to 3.141595, or even shorter, 3.14.
The area formula is really the formula for the curved surface (that is the portion) added to the area of both ends (that is the portion). If you see the phrase, "area of cylinder base," the writer is referring to the top and bottom ends, not the curved surface between them.
The area will always be expressed in square units stemming from the linear units in the problem, since any two linear measures multiplied by one another yield square units.
Lateral surface area of a cylinder
As mentioned above, there is also the lateral surface area of an object. The lateral surface of an object is defined as the area of all the sides of the object, excluding the area of its base and top. For a cylinder, the lateral surface area is the curved surface that connects the base and the top.
Lateral surface area formula
The formula for calculating the lateral surface area is similar to the surface area formula above, but since we are not including the top or base, we must remove that part of the formula. The lateral surface area formula is:
How to find the surface area of a cylinder
Take the cardboard cylinder from the core of a roll of paper towels. Imagine it is 1.7-inch diameter, 11-inch long.
Knowing the cardboard tube is 1.7 inches wide and 11 inches long, if we were to close up the ends, what would be its surface area?
Here we did not have the radius, r; we were told the diameter, but recalling the relationship between parts of circles, we remember:
Put the values of r and h into the formula, using 3.14 for π:
A paper towel tube uses slightly more than 0.4 square feet of cardboard!
Approximately how much aluminum do you estimate is needed to make a typical soda can?
True, a soda can dents in a bit at both ends, but we just need an estimate. First, we need the radius, so we take half of the can's diameter:
Let's recall our formula and plug in what we know:
That is a little more than a third of a square foot of recycled aluminum in every can. Now you can see why recycling aluminum is important - just three cans uses up a square foot of the metal!