Right Cylinder

Q: How do you find the height of a right cylinder given its volume?

Rearrange the volume formula to solve for height: $h = \frac{V}{B}$. If the base is a circle with radius $r$, this becomes $h = \frac{V}{\pi r^2}$. Simply divide the volume by the base area.

Right Cylinder

A cylinder which has bases aligned one directly above the other. The bases need not be circles.

Right cylinder diagram with height h and base radius, showing formulas: Volume=Bh, Lateral Surface Area=hC, Total Surface...

Key Formula

V = B \cdot h \qquad \text{and} \qquad SA = 2B + P \cdot h

Where:

$V$ = Volume of the right cylinder
$B$ = Area of one base
$h$ = Height (altitude) of the cylinder — the perpendicular distance between the two bases
$SA$ = Total surface area of the right cylinder
$P$ = Perimeter (or circumference) of one base

Worked Example

Problem: A right circular cylinder has a base radius of 5 cm and a height of 12 cm. Find its volume and total surface area.

Step 1: Find the area of one circular base.

B = \pi r^2 = \pi (5)^2 = 25\pi \approx 78.54 \text{ cm}^2

Step 2: Find the circumference (perimeter) of the base.

P = 2\pi r = 2\pi(5) = 10\pi \approx 31.42 \text{ cm}

Step 3: Calculate the volume using V = B · h.

V = 25\pi \cdot 12 = 300\pi \approx 942.48 \text{ cm}^3

Step 4: Calculate the lateral surface area using P · h.

\text{Lateral Area} = 10\pi \cdot 12 = 120\pi \approx 376.99 \text{ cm}^2

Step 5: Calculate total surface area by adding the two bases to the lateral area.

SA = 2(25\pi) + 120\pi = 50\pi + 120\pi = 170\pi \approx 534.07 \text{ cm}^2

Answer: The volume is

300\pi \approx 942.48

cm³ and the total surface area is

170\pi \approx 534.07

cm².

Another Example

This example uses a non-circular base (a regular hexagon) to show that the definition of a right cylinder is not limited to circles. The same general formula V = B · h applies regardless of the base shape.

Problem: A right cylinder has a regular hexagonal base with side length 4 cm and a height of 10 cm. Find the volume of this cylinder.

Step 1: Recall the area formula for a regular hexagon with side length s.

B = \frac{3\sqrt{3}}{2}\,s^2

Step 2: Substitute s = 4 cm to find the base area.

B = \frac{3\sqrt{3}}{2}(4)^2 = \frac{3\sqrt{3}}{2} \cdot 16 = 24\sqrt{3} \approx 41.57 \text{ cm}^2

Step 3: Multiply the base area by the height to find the volume.

V = B \cdot h = 24\sqrt{3} \cdot 10 = 240\sqrt{3} \approx 415.69 \text{ cm}^3

Answer: The volume is

240\sqrt{3} \approx 415.69

cm³.

Frequently Asked Questions

What is the difference between a right cylinder and an oblique cylinder?

In a right cylinder, the axis (the line connecting the centers of the two bases) is perpendicular to the bases, so the bases sit directly above one another. In an oblique cylinder, the axis is tilted at an angle, making the shape lean to one side. Both have the same volume formula V = Bh, but the lateral surface area calculation differs because the slant changes in an oblique cylinder.

Does a right cylinder always have circular bases?

No. A right cylinder can have any congruent, parallel bases — circles, ellipses, or even polygons. When the bases are specifically circles, the shape is called a right circular cylinder, which is the most common type you encounter in textbooks.

How do you find the height of a right cylinder given its volume?

Rearrange the volume formula to solve for height:

h = \frac{V}{B}

. If the base is a circle with radius

r

, this becomes

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

. Simply divide the volume by the base area.

Right Cylinder vs. Oblique Cylinder

	Right Cylinder	Oblique Cylinder
Axis orientation	Perpendicular to the bases	Tilted at an angle to the bases
Volume formula	V = B · h	V = B · h (same — height is still the perpendicular distance)
Lateral surface area (circular base)	2πrh	2πr · l, where l is the slant height (l > h)
Cross-section parallel to base	Identical to the base at every height	Identical to the base at every height
Typical appearance	Stands straight up like a soup can	Leans to one side like a leaning stack of coins

Why It Matters

Right cylinders appear throughout geometry courses when studying volume and surface area of three-dimensional solids. In real life, most cans, pipes, and pillars are right circular cylinders, making these formulas essential for engineering and design problems. Understanding the general right cylinder (with non-circular bases) also prepares you for prisms, since a prism is technically a right cylinder with polygonal bases.

Common Mistakes

Mistake: Assuming a right cylinder must have circular bases.

Correction: The word 'right' refers to the axis being perpendicular to the bases, not the base shape. Any congruent, parallel bases work. If you specifically need circular bases, the correct term is right circular cylinder.

Mistake: Using slant height instead of perpendicular height in the volume formula.

Correction: For volume, always use the perpendicular distance between the bases (the altitude h). Slant height is only relevant for lateral surface area of oblique cylinders. In a right cylinder, the height and the axis length are the same, so this error is less likely — but be careful when a problem gives both measurements.

Related Terms

Cylinder — General parent category of all cylinders
Right Circular Cylinder — A right cylinder specifically with circular bases
Oblique Cylinder — A cylinder with a tilted (non-perpendicular) axis
Base — The congruent parallel faces of the cylinder
Altitude of a Cylinder — The perpendicular height h used in formulas
Lateral Surface Area — Area of the curved side surface (P · h)
Volume — Space enclosed, calculated as B · h
Surface Area — Total area including bases and lateral surface