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Oblique

Oblique

Tilted at an angle; neither vertical nor horizontal.

 

 

 

See also

Oblique asymptote, oblique cone, oblique cylinder, oblique prism, oblique pyramid

Worked Example

Problem: A cylinder has a circular base with radius 5 cm. Instead of standing straight up, its top face is shifted 4 cm to the right while remaining at a height of 10 cm. Is this cylinder oblique or right, and what is the length of its lateral (slant) edge?
Step 1: Determine whether the cylinder is oblique. The axis connecting the centers of the two circular faces is not perpendicular to the base — the top center is offset 4 cm horizontally. Since the axis is tilted, this is an oblique cylinder.
Step 2: Find the slant length of the axis. The vertical distance between the bases is 10 cm and the horizontal offset is 4 cm. Use the Pythagorean theorem to find the axis length.
ℓ=102+42=100+16=116≈10.77 cm\ell = \sqrt{10^2 + 4^2} = \sqrt{100 + 16} = \sqrt{116} \approx 10.77 \text{ cm}
Step 3: Note that the volume of the oblique cylinder is the same as a right cylinder with the same base and perpendicular height (by Cavalieri's principle).
V=πr2h=π(5)2(10)=250π≈785.4 cm3V = \pi r^2 h = \pi (5)^2 (10) = 250\pi \approx 785.4 \text{ cm}^3
Answer: The cylinder is oblique. Its axis has a slant length of approximately 10.77 cm, and its volume is 250π≈785.4250\pi \approx 785.4 cm³.

Another Example

Problem: Two lines cross so that one of the angles formed is 60°. Are the lines oblique to each other?
Step 1: Check whether the lines meet at a right angle. A 60° angle is not 90°, so the lines are not perpendicular.
Step 2: Check whether the lines are parallel. They cross, so they are not parallel either.
Step 3: Since the lines intersect at an angle that is neither 0° (parallel) nor 90° (perpendicular), they are oblique to each other.
Answer: Yes, the lines are oblique because they meet at 60°, which is neither parallel nor perpendicular.

Frequently Asked Questions

What is the difference between oblique and perpendicular?
Perpendicular means two lines or surfaces meet at exactly 90°. Oblique means they meet at any angle other than 90°. Both involve intersection, but only perpendicular implies a right angle.
Does an oblique shape have a different volume than a right shape?
For prisms, cylinders, and pyramids, tilting the solid does not change the volume as long as the base area and perpendicular height stay the same. This is guaranteed by Cavalieri's principle. However, the surface area of an oblique solid is generally different from its right counterpart.

Right (solid or angle) vs. Oblique (solid or angle)

A 'right' solid has its axis perpendicular to the base — for example, a right cylinder stands straight up. An 'oblique' solid has its axis tilted so it is not perpendicular to the base. Similarly, when two lines meet at 90° they form a right angle; any other non-zero crossing angle is called oblique. The key distinction is always whether the relationship involves a 90° angle (right) or not (oblique).

Why It Matters

The oblique vs. right distinction appears throughout geometry whenever you compute surface area, because tilting a solid changes which faces are congruent and how you measure slant height. In trigonometry, oblique triangles (those without a right angle) require the law of sines or the law of cosines instead of simpler right-triangle ratios. Understanding the concept also helps in calculus, where oblique asymptotes describe the end behavior of rational functions.

Common Mistakes

Mistake: Assuming an oblique prism or cylinder has a different volume than the corresponding right version.
Correction: Volume depends on base area and perpendicular height, not slant height. By Cavalieri's principle, tilting a solid does not change its volume. Use the perpendicular height, not the slant length, in volume formulas.
Mistake: Confusing 'oblique' with 'obtuse' when describing angles.
Correction: 'Oblique' means any angle that is not 0° or 90° — it could be acute (less than 90°) or obtuse (greater than 90°). 'Obtuse' specifically means an angle between 90° and 180°.

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