Altitude of a Cylinder

Q: How do you find the altitude of a cylinder if you know the volume and radius?

Rearrange the volume formula: $h = \dfrac{V}{\pi r^2}$. For example, if the volume is $200\pi$ cm³ and the radius is 5 cm, then $h = \dfrac{200\pi}{\pi \cdot 25} = 8$ cm.

Altitude of a Cylinder
Height of a Cylinder

The distance between the bases of a cylinder. Formally, the shortest line segment between the (possibly extended) bases. Altitude also refers to the length of this segment.

Key Formula

V = \pi r^2 h

Where:

$V$ = Volume of the cylinder
$r$ = Radius of the circular base
$h$ = Altitude (height) of the cylinder — the perpendicular distance between the bases

Worked Example

Problem: A right cylinder has a base radius of 5 cm and an altitude of 12 cm. Find its volume.

Step 1: Identify the altitude. The altitude is the perpendicular distance between the two circular bases, which is given as 12 cm.

h = 12 \text{ cm}

Step 2: Write the volume formula for a cylinder.

V = \pi r^2 h

Step 3: Substitute the radius and altitude into the formula.

V = \pi (5)^2 (12) = \pi \cdot 25 \cdot 12

Step 4: Compute the result.

V = 300\pi \approx 942.48 \text{ cm}^3

Answer: The volume of the cylinder is

300\pi \approx 942.48

cm³.

Another Example

Problem: An oblique cylinder has circular bases of radius 4 cm. The lateral side makes a slant length of 10 cm, but the perpendicular distance between the planes of the two bases is 8 cm. What is the altitude, and what is the volume?

Step 1: Distinguish the altitude from the slant length. The altitude is the perpendicular (shortest) distance between the two base planes, not the slant length along the side.

h = 8 \text{ cm (not 10 cm)}

Step 2: Use the volume formula. Even for an oblique cylinder, volume depends on the altitude, not the slant length.

V = \pi r^2 h = \pi (4)^2 (8)

Step 3: Calculate.

V = 128\pi \approx 402.12 \text{ cm}^3

Answer: The altitude is 8 cm, and the volume is

128\pi \approx 402.12

cm³.

Frequently Asked Questions

Is the altitude of a cylinder the same as its height?

Yes. The terms 'altitude' and 'height' are used interchangeably for a cylinder. Both refer to the perpendicular distance between the two bases. The word 'altitude' emphasizes that the measurement is perpendicular, which matters especially for oblique cylinders.

How do you find the altitude of a cylinder if you know the volume and radius?

Rearrange the volume formula:

h = \dfrac{V}{\pi r^2}

. For example, if the volume is

200\pi

cm³ and the radius is 5 cm, then

h = \dfrac{200\pi}{\pi \cdot 25} = 8

cm.

Altitude of a cylinder vs. Slant height of an oblique cylinder

The altitude is always the perpendicular distance between the two base planes. The slant height is the length measured along the tilted lateral surface from one base to the other. For a right cylinder, the altitude and the lateral edge length are equal because the side is perpendicular to the bases. For an oblique cylinder, the slant length is longer than the altitude. Volume always uses the altitude, never the slant height.

Why It Matters

The altitude appears in every major cylinder calculation. Volume equals the base area times the altitude (

V = \pi r^2 h

), and the lateral surface area of a right cylinder is the circumference times the altitude (

A_{\text{lateral}} = 2\pi r h

). Understanding that altitude means the perpendicular distance is essential when working with oblique cylinders, where a careless measurement along the slanted side gives the wrong answer.

Common Mistakes

Mistake: Using the slant length instead of the altitude for an oblique cylinder's volume.

Correction: The altitude is the perpendicular distance between the base planes. In an oblique cylinder, the slant side is longer than the altitude. Always use the perpendicular distance in the volume formula.

Mistake: Confusing the altitude with the radius or the diameter.

Correction: The radius and diameter measure across the circular base. The altitude measures the distance between the two bases, which is a completely different dimension.

Related Terms

Cylinder — The solid whose altitude is being measured
Right Cylinder — Altitude equals the lateral edge length
Oblique Cylinder — Altitude differs from the slant length
Base — The two parallel faces between which altitude is measured
Altitude — General term for perpendicular height
Volume — Calculated using the altitude of the cylinder
Surface Area — Lateral area uses the altitude for right cylinders
Line Segment — The altitude is a specific line segment