Axis of a Cylinder
Worked Example
Problem: A right circular cylinder has its bottom base centered at the point (0, 0, 0) and its top base centered at the point (0, 0, 8). The radius of each base is 5. Describe the axis of the cylinder and find its length.
Step 1: Identify the centers of the two bases. The bottom base is centered at (0, 0, 0) and the top base is centered at (0, 0, 8).
C1=(0,0,0),C2=(0,0,8)
Step 2: The axis is the line segment connecting these two centers. Since only the z-coordinate changes, the axis runs straight up along the z-axis.
Axis: {(0,0,z)∣0≤z≤8}
Step 3: Find the length of the axis using the distance formula between the two centers.
d=(0−0)2+(0−0)2+(8−0)2=64=8
Step 4: Because the axis is perpendicular to both bases, this cylinder is a right cylinder. The length of the axis equals the height of the cylinder.
Answer: The axis is the vertical line segment from (0, 0, 0) to (0, 0, 8), with a length of 8 units. This length equals the height of the cylinder.
Frequently Asked Questions
Is the axis of a cylinder always equal to its height?
For a right cylinder, yes — the axis is perpendicular to the bases, so its length equals the height. For an oblique cylinder, the axis is tilted, so the axis length is greater than the perpendicular height. The height of a cylinder is always the perpendicular distance between the two bases, regardless of the axis orientation.
Does every cylinder have an axis?
Yes. Every cylinder has two parallel, congruent circular bases, and each base has a center. The line connecting those two centers is always the axis. This applies to both right and oblique cylinders.
Axis of a right cylinder vs. Axis of an oblique cylinder
In a right cylinder, the axis is perpendicular to the bases, so axis length equals the height. In an oblique cylinder, the axis is slanted relative to the bases, so the axis is longer than the perpendicular height.
Why It Matters
The axis of a cylinder is central to calculating volume and surface area, since the height used in those formulas is measured along (or parallel to) the axis for a right cylinder. In manufacturing and engineering, the axis defines the line around which a cylindrical object is symmetric, which is essential for tasks like drilling, turning on a lathe, or aligning pipes. Understanding the axis also helps you distinguish between right and oblique cylinders, which behave differently in geometry problems.
Common Mistakes
Mistake: Confusing the axis length with the height for an oblique cylinder.
Correction: The height of a cylinder is always the perpendicular distance between the two bases. In an oblique cylinder, the axis is slanted, so its length is greater than the height. Use the perpendicular height — not the axis length — in volume formulas.
Mistake: Thinking the axis is a surface feature or an edge of the cylinder.
Correction: The axis is an imaginary line running through the interior of the cylinder, connecting the centers of the two bases. It is not a visible edge or part of the lateral surface.
Related Terms
- Cylinder — The solid whose axis is defined
- Base — The circular ends whose centers define the axis
- Line — The axis is a line segment
- Right Cylinder — Cylinder whose axis is perpendicular to bases
- Oblique Cylinder — Cylinder whose axis is tilted from perpendicular
- Axis of a Cone — Analogous axis concept for cones
- Height — Perpendicular distance between bases