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Right Circular Cylinder — Definition, Formula & Examples

Right Circular Cylinder

A right cylinder with bases that are circles.

 

Cylinder with height h and base radius r. Formulas: Volume=πr²h, Lateral Surface Area=2πrh, Total Surface Area=2πr²+2πrh

 

 

See also

Cylinder, circular cylinder, height of a cylinder, volume, lateral surface, lateral surface area, surface area

Key Formula

V=πr2hAlateral=2πrhAtotal=2πrh+2πr2V = \pi r^2 h \qquad A_{\text{lateral}} = 2\pi r h \qquad A_{\text{total}} = 2\pi r h + 2\pi r^2
Where:
  • VV = Volume of the cylinder
  • AlateralA_{\text{lateral}} = Lateral (side) surface area of the cylinder
  • AtotalA_{\text{total}} = Total surface area, including both circular bases
  • rr = Radius of the circular base
  • hh = Height (altitude) of the cylinder, measured perpendicular to the bases
  • π\pi = The constant pi, approximately 3.14159

Worked Example

Problem: A right circular cylinder has a base radius of 5 cm and a height of 12 cm. Find the volume, lateral surface area, and total surface area.
Step 1: Find the volume using the formula V=πr2hV = \pi r^2 h.
V=π(5)2(12)=π⋅25⋅12=300π≈942.5 cm3V = \pi (5)^2 (12) = \pi \cdot 25 \cdot 12 = 300\pi \approx 942.5 \text{ cm}^3
Step 2: Find the lateral surface area using Alateral=2πrhA_{\text{lateral}} = 2\pi r h.
Alateral=2π(5)(12)=120π≈376.99 cm2A_{\text{lateral}} = 2\pi (5)(12) = 120\pi \approx 376.99 \text{ cm}^2
Step 3: Find the area of the two circular bases combined.
Abases=2πr2=2π(5)2=50π≈157.08 cm2A_{\text{bases}} = 2\pi r^2 = 2\pi (5)^2 = 50\pi \approx 157.08 \text{ cm}^2
Step 4: Add the lateral area and the base areas to get the total surface area.
Atotal=120π+50π=170π≈534.07 cm2A_{\text{total}} = 120\pi + 50\pi = 170\pi \approx 534.07 \text{ cm}^2
Answer: Volume =300π≈942.5= 300\pi \approx 942.5 cm³, Lateral surface area =120π≈377.0= 120\pi \approx 377.0 cm², Total surface area =170π≈534.1= 170\pi \approx 534.1 cm².

Another Example

This example works backward from a given volume to find the radius first, then uses that radius to compute a surface area — a common variation in real-world and exam problems.

Problem: A cylindrical water tank holds exactly 500π cubic feet of water. If the tank's height is 20 feet, find the radius of the base and the amount of material needed to construct just the lateral (side) surface.
Step 1: Start with the volume formula and solve for rr.
V=πr2h  ⟹  500π=πr2(20)V = \pi r^2 h \implies 500\pi = \pi r^2 (20)
Step 2: Divide both sides by 20π20\pi to isolate r2r^2.
r2=500π20π=25  ⟹  r=5 ftr^2 = \frac{500\pi}{20\pi} = 25 \implies r = 5 \text{ ft}
Step 3: Now compute the lateral surface area using the radius you found.
Alateral=2π(5)(20)=200π≈628.3 ft2A_{\text{lateral}} = 2\pi (5)(20) = 200\pi \approx 628.3 \text{ ft}^2
Answer: The base radius is 5 ft, and the lateral surface area is 200π≈628.3200\pi \approx 628.3 ft².

Frequently Asked Questions

What is the difference between a cylinder and a right circular cylinder?
"Cylinder" is a general term that can refer to shapes with non-circular cross-sections (elliptical cylinders) or axes that are not perpendicular to the base (oblique cylinders). A right circular cylinder specifically has circular bases and an axis that is perpendicular to those bases. In most textbooks, when they say "cylinder" without qualification, they mean a right circular cylinder.
Why is the lateral surface area of a right circular cylinder equal to 2πrh?
If you "unroll" the curved side surface, it flattens into a rectangle. The rectangle's width equals the circumference of the base circle, which is 2πr2\pi r, and its height equals the cylinder's height hh. The area of that rectangle is therefore 2πr×h2\pi r \times h.
How do you find the height of a right circular cylinder given the volume and radius?
Rearrange the volume formula V=πr2hV = \pi r^2 h to solve for hh: divide both sides by πr2\pi r^2 to get h=Vπr2h = \frac{V}{\pi r^2}. Plug in the known values for VV and rr to calculate the height.

Right Circular Cylinder vs. Oblique Circular Cylinder

Right Circular CylinderOblique Circular Cylinder
Axis orientationAxis is perpendicular to the basesAxis is tilted at an angle to the bases
Cross-section shapeEvery horizontal cross-section is a circle congruent to the baseHorizontal cross-sections are circles, but the axis does not pass through their centers vertically
Volume formulaV=πr2hV = \pi r^2 hV=πr2hV = \pi r^2 h (same, where hh is the perpendicular height, not the slant)
Lateral surface area2πrh2\pi r h — straightforward rectangle when unrolledMore complex; cannot be unrolled into a simple rectangle
Common usageStandard cylinder in most textbooks and applicationsLess common; appears in advanced geometry problems

Why It Matters

Right circular cylinders appear constantly in geometry courses, standardized tests (SAT, ACT), and physics problems involving fluid volume, pressure, and heat transfer. Engineering and manufacturing rely on cylinder calculations for designing pipes, tanks, pistons, and containers. Understanding this shape also prepares you for calculus topics like solids of revolution and integration in cylindrical coordinates.

Common Mistakes

Mistake: Using the diameter instead of the radius in the formulas.
Correction: The formulas V=πr2hV = \pi r^2 h and A=2πrhA = 2\pi r h require the radius, which is half the diameter. If a problem gives you a diameter of 10, use r=5r = 5. Forgetting to halve the diameter will make your volume four times too large.
Mistake: Forgetting to include both circular bases when calculating total surface area.
Correction: The total surface area formula is Atotal=2πrh+2πr2A_{\text{total}} = 2\pi r h + 2\pi r^2. The term 2πr22\pi r^2 accounts for the top and bottom circles. If you only add one base area (πr2\pi r^2), your answer will be too small. Note that some real-world problems (like an open-top tank) intentionally exclude one base — read carefully.

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