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Understanding the difference between volume and surface area is important in mathematics and practical applications like gift wrapping. Volume is the capacity of an object, while surface area is the sum of all its faces. Calculating volume is straightforward, but surface area requires finding the area of each face. Calculating the surface area of a cylinder is more complex, involving the lateral area and the areas of the circular faces. These concepts are useful in fields like construction and engineering.
Volume and surface area are two related concepts in the study of mathematics. They are both important to understand, but just as important is understanding how they differ and what they mean. This is especially true when it comes to calculating the volume and surface area of a prism or cylinder.
If you plan on wrapping a gift in a box, you can get an idea of how the volume and surface area differ. First, you need to consider the size of the box, when considering the size of the gift. How much internal space does your box need to fit a gift? The measure of the box’s capacity, how much it will hold, is its volume. Then you have to wrap the gift. The amount of wrapping paper that will cover the outside of the box is a very different calculation than the capacity of the box. You’ll need a separate measurement, or some good guess, to calculate the sum of the sides of all surfaces or the surface area.
The volume of a square or rectangular box is easy enough to calculate. Simply multiply the height by the length by the width to get your measurement. With a square it’s even easier, just make a cube the length of one side, as they all measure the same. If the side length is a, the formula is axaxa or a3. When you compare volume and surface area, you’ll notice a very different formula. You need to get the area of each face and then add the areas of all faces together. With a square prism or cube, you would essentially calculate the area axa or a2, multiplied by 6 (6a2). When working with a rectangular prism, you’ll need to come up with the area of 3 pairs of equal sides, which must be added together to determine the surface area.
The work on volume and surface area differs slightly when trying to calculate the area of a cylinder. The formula for a volume of a cylinder is the area of a circular face multiplied by the height of the cylinder. It reads: πr2 xh, or pi times radius squared times height. Getting the surface area of the cylinder is a little more complicated since the circular portion is essentially one continuous face. Calculating the surface area of a cylinder means calculating the lateral area of this face.
The lateral area formula is as follows πr2r or πd (pi times the radius doubled or pi times the diameter), multiplied by the height, πr2r x h. This is essentially the circumference of a circle times the height of the cylinder. To calculate the whole formula, the areas of the upper and lower circular faces must also be added. Since they are equal in a cylinder, the formula is 2 πr2. This calculation is then added to the lateral area to calculate the full surface area in the following expression:
πr2r xh + 2πr2 = area laterale.
You can also view the difference between volume and cylinder as the difference between what is inside and can be contained and the outside of a three-dimensional object. These are invaluable differences to understand in many applications, such as construction, engineering, or even gift wrapping. When kids complain that math is useless outside of math class, you might point out that knowing the difference between volume and surface area meant they got a very well-wrapped gift on their birthday.
