The next 3D shape we shall look at is the circular cylinder (also called the “right circular cylinder” by the GRE for its right angles):
Circular Cylinder: A 3D shape that has a circle for both bases and a perpendicular line connecting the center of those bases.
Now, it follows from the definition that a circular cylinder’s top and bottom circles will be the same size. So we use to denote the radius of either circle, and we use to denote the height of the cylinder. Without further ado, we present:
Volume of a Circular Cylinder: The volume of a right circular cylinder with a radius of and a height of is
To get the area of the rectangle, you can imagine taking the base which is units long:
and extending it upwards by multiplying by . Thus, we get that the area is
Similarly, in our above formula, you can imagine that we are taking the base of the cylinder (which has an area of , following our formula for the area of a circle) and extending that base upwards (so multiplying by ) in order to get
Surface Area of a Circular Cylinder: The surface area of a circular cylinder with a radius of and a height of is
Now, there is no mystery about where we get the term. That just comes from the area of the top circle plus the area of the bottom circle. Now, to finish getting the surface area, we just need to find some way to capture the area of the curved portion between the two ends.
Now, imagine the cylinder was in front of you. You can picture an empty can of soda, if that helps. Imagine cutting off the top and bottom circles, so that you just have the curved portion in your hands. Now suppose you cut one side of that surface and then unwrap the curved portion. Lay it flat against a table. What would you get?
Yes, you'd get a rectangle. Now, we know how to find the area of a rectangle. You just multiply the two sides together! And we know the bottom side, since that's just the height of our original cylinder. So we just want to find the left-hand side of this rectangle.
Now, originally, this rectangle was wrapped around the circumference of the circles at the bottom and top of our cylinder. So the base of the rectangle is just equal to the circumference of those circles, which is We get:
Thus, the area of the rectangle is And we add that to the of the circles to get the total surface area of our cylinder.
1. The cylinder below has a radius of and a height of 6. Find .
The key here is to note that is the hypotenuse of a right triangle with legs at and And is just the radius of one of the circles, so it equals while is just the height of the cylinder, or in other words, 6. Thus,
2. The surface area of a right circular cylinder is and its radius is 2. What is the height of the cylinder?
Let's use our formula for the surface area of such a cylinder:
Thus, and the height of the cylinder is 4 units.
3. The surface area of a cylinder is twice its volume. Its radius equals its height. What is the volume of the cylinder?
Let's say that the radius of the cylinder is and its height is . Then, the question tells us that:
And replacing with , we get:
And so and the volume of the cylinder is