The total surface of a geometric solid is given by the sum of the area of each of the faces that compose it. To calculate the area occupied by the surface of a cylinder, it is necessary to calculate the area of the two bases and add it to the area of the cylindrical section between them. The mathematical formula for calculating the area of a cylinder is A = 2 π r2 + 2 π r h.
Steps
Part 1 of 3: Calculate the Area of the Bases
Step 1. Mentally visualize the top and bottom of a cylinder
If you can't, you can use any food can - they all have a cylindrical shape. Looking at any cylindrical object you will notice that the upper and lower bases are the same and have a circular shape. The first step in calculating the surface of a cylinder therefore consists in calculating the area of the two circular bases that delimit it.
Step 2. Find the radius of the cylinder under consideration
The radius is the distance between the center of a circle and any point on the circumference. The mathematical sign that identifies the radius is "r". In the case of a cylinder, the radius of the two bases is always the same. In our example we assume that we have a cylinder with a radius of 3 cm.
- If you are taking a math exam or are doing your school assignments, the value of the radius should be clearly expressed in the text of the problem to be solved. The diameter value should also be known. The diameter of a circle is the measurement of the segment passing through the center that joins two points on the circumference. The radius of a circle is exactly half the diameter.
- If you need to calculate the area of a real cylinder, you can measure its radius using a simple ruler.
Step 3. Calculate the area of the upper base
The area of a circle is given by the product of the constant π (whose rounded value is equal to 3.14) and the square of the radius. The mathematical formula is the following: A = π * r2. Simplifying it further we can use this formula: A = π * r * r.
- To calculate the area of the base of the cylinder under consideration, simply substitute A = πr in the formula2, the value of the radius, which in our example is equal to 3 cm. By carrying out the calculations we will obtain:
- A = π * r2
- A = π * 32
- A = π * 9 = 28.26 cm2
Step 4. Repeat the procedure to calculate the area of the second base
Now that we have calculated the area of the upper base of the cylinder, it is necessary to take into account that the lower base also exists. To calculate the area of the latter, you can perform the calculations described in the previous step again or, since the two bases are identical, you can simply double the value already obtained.
Part 2 of 3: Calculate the Side Surface Area of the Cylinder
Step 1. Mentally visualize the section of a cylinder between the two bases
When you look at a can of beans, you can easily spot the top and bottom base. These two "faces" of the solid are connected to each other by a circular section (represented by the body of our can of beans). The radius of the cylindrical section is identical to that of the two bases, but we will also have to take into account its height.
Step 2. Calculate the circumference of the cylinder under consideration
To calculate the side surface area of our cylinder, we must first calculate its circumference. To do this, simply multiply the radius by the constant π and double the result. Using the data in our possession we will obtain: 3 * 2 * π = 18, 84 cm.
Step 3. Multiply the circumference by the height of the cylinder
This will give you the side surface area of the solid. Then proceed by multiplying the circumference, equal to 18.84 cm, by the height, which we assume to be 5 cm. Using the given formula we will get: 18, 84 * 5 = 94, 2 cm2.
Part 3 of 3: Calculating the Total Area of a Cylinder
Step 1. View the entire cylinder
The first step was to obtain the area of the two bases and then proceed to calculate the area of the lateral surface of the solid between them. At this point, you have to visualize the solid in its entirety (with the help of our can of beans) and proceed to calculate the total surface.
Step 2. Double the area of a single base
To do this, simply multiply by 2 the value obtained in the first part of the article: 28, 26 cm2. Carrying out the calculation you will get: 28.26 * 2 = 56.52 cm2. Now you have the area of both bases that make up the cylinder.
Step 3. Add the area of the bases to that of the side surface of the cylinder
In this way you will get the total surface area of the cylinder under examination. The calculations are very simple, you need to add 56.52 cm2, ie the total area of the two bases, at 94.2 cm2. By performing the calculation you will get: 56, 52 cm2 + 94, 2 cm2 = 150, 72 cm2. We can conclude that the total area of a cylinder 5 cm high and having a circular base of 3 cm in radius is equal to 150, 72 cm2.
