Volume measures the capacity of an object.
Volume and surface area are two related concepts in the study of mathematics. Both are important to understand, but equally important is understanding how they are different and what they mean. This is especially the case when it comes to calculating the volume and surface areas of a prism or cylinder.
If you think about wrapping a gift in a box, you can get a good idea of how volume and surface area differ. First, you need to consider the size of the box when considering the size of the gift. How much interior space does your box need to fit a gift? The measure of the box’s capacity, how much it can 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 will need a separate measurement or some good guesses to calculate the sum of the sides of all the surfaces, or the area of the surface.
The volume of a square or rectangular box is very easy to calculate. Simply multiply the height times the length and width to get the measurement. With a square it is even easier, just cube the length of one side, since they all measure the same. If the side length is a , the formula is axaxa or 3 . 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 the faces. With a square or cubic prism, you would basically calculate the area axa or a 2 , multiplied by 6 (6a 2 ). When you work with a rectangular prism, you will have the area of 3 pairs of equal sides, which must be added to determine the surface area.
Working on volume and surface area is a bit different when trying to calculate the area of a cylinder. The formula for the volume of a cylinder is the area of a circular face multiplied by the height of the cylinder. It says: πr 2 xh, or pi times the radius squared times the height. Obtaining the surface area of the cylinder is a bit more complicated, since the circular portion is essentially a continuous face. Calculating the surface area of a cylinder means calculating the lateral area of this face.
The formula for lateral area is as follows: πr2r or πd (pi times the fold radius 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 full formula, you must also add the areas of the top and bottom circular faces. Since they are equal in a cylinder, the formula is 2 πr 2 . This calculation is then added to the lateral area to calculate the total surface area in the following expression:
πr2r xh + 2πr 2 = lateral area.
You can also see the difference between volume and cylinder as a difference between what is inside and can be contained and what is outside of a three-dimensional object. These are valuable 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 can tell them that knowing the difference between volume and surface area means they got a beautifully wrapped birthday present.