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4.6 E) Cylinders
4.6 E) Cylinders
A cylinder is very similar to a prism because the two ends of the cylinder are the same; they are both circles. Also, if you were to cut the cylinder along its length, the cross section will always be a circle. However, a cylinder is not a prism because it has a curved face and all of the faces on a prism are flat.
A cylinder has 3 faces; two circular cross sections (the top and the bottom) and a curved surface.
We work out the volume of a cylinder in the same way that we worked out the volume of a prism. The only difference is that we refer to the length as the height of the cylinder. The formula is:
We work out the volume of a cylinder in the same way that we worked out the volume of a prism. The only difference is that we refer to the length as the height of the cylinder. The formula is:
The cross section of a cylinder is a circle. This means that we can replace the area of cross section with the area of a circle formula.
We can tidy up this formula and let h be the height.
Let’s have an example.
Example 1
What is the volume of the cylinder below? Give you answer to one decimal place.
What is the volume of the cylinder below? Give you answer to one decimal place.
From the diagram, we can see that we are given the height of the cylinder and the diameter of the circular cross section. We work out the volume of a cylinder by using the formula below:
In order to work out the volume of the cylinder, we need the height of the cylinder and the radius of the circular cross section. Currently, we have the height of the cylinder (12 cm), but we do not have the radius of the circular cross section. However, we do have the diameter of the cross section (8 cm) and we know that the radius is half the length of the diameter. This means that the radius is 4 cm (8 ÷ 2).
We can sub these values into the formula:
We can sub these values into the formula:
We are asked in the question to give our answer to one decimal place.
The volume of the cylinder is 603.2 cm3.
Sometimes it will be the case that you are asked to give you answer in exact form and you do this by leaving the answer in terms of π.
Surface Area
The working out of the surface area of a cylinder is fairly tricky. This is because a cylinder has 3 different faces; the two circular ends (top and bottom) and a curved surface. The curved surface can be flattened out to make a rectangular shape, with one side being the height of the cylinder and the other side being the circumference of the circular cross section. To work out the area of this curved face, we multiply the circumference of the circle by the height of the cylinder. We work out the circumference of the circle by either using 2πr (where r is the radius) or πd (where d is the diameter).
The working out of the surface area of a cylinder is fairly tricky. This is because a cylinder has 3 different faces; the two circular ends (top and bottom) and a curved surface. The curved surface can be flattened out to make a rectangular shape, with one side being the height of the cylinder and the other side being the circumference of the circular cross section. To work out the area of this curved face, we multiply the circumference of the circle by the height of the cylinder. We work out the circumference of the circle by either using 2πr (where r is the radius) or πd (where d is the diameter).
The total surface area of a cylinder is given by the formula below.
Or:
The first term of both of the formulas works out the area of the two cross sections. The second term in both of the formulas works out the area of the curve surface.
It does not matter which formula you use as both of them will get you to the same answer. I think that it is best to use the first formula because the first formula just contains the radius (whereas the second formula contains both the radius and the diameter).
Let’s use the formula to work out the surface area of the cylinder that was given in the first example (to one decimal place). A diagram of the cylinder is shown below:
It does not matter which formula you use as both of them will get you to the same answer. I think that it is best to use the first formula because the first formula just contains the radius (whereas the second formula contains both the radius and the diameter).
Let’s use the formula to work out the surface area of the cylinder that was given in the first example (to one decimal place). A diagram of the cylinder is shown below:
All we need to do to work out the surface area of the cylinder is to sub the respective values into the correct places in the formula. I am going to use the first formula, which is shown below:
In order to use the above formula, we need to know the height of the cylinder and the radius of the circular cross section. The height of the cylinder is 12 cm. We do not know what the radius of the circular cross section is. However, we are told that the diameter of the circular cross section is 8 cm. The radius is half the length of the diameter, which means that the radius of the circular cross section is 4 cm (8 ÷ 2).
We can now sub the height and radius into the formula:
We can now sub the height and radius into the formula:
We now give our answer to 1 decimal place.
The total surface area of the cylinder is 402.1 cm2.