# Right Circular Cylinder: Definition, Volume, Surface Area, and Examples

TL;DR

A right circular cylinder is a 3D solid with two equal, parallel circular bases joined by a curved surface, where the axis joining the base centres stands perpendicular to the bases. Its volume is \(\pi r^2 h\), its curved surface area is \(2\pi rh\), and its total surface area is \(2\pi r(h + r)\). This article derives each formula, explains exactly what makes a cylinder "right circular" rather than oblique, and works through examples and the slips students hit most.

## What is a Right Circular Cylinder?

A **right circular cylinder** is a three-dimensional solid with **two equal, parallel circular bases** joined by a single **curved (lateral) surface**, where the line joining the centres of the two bases — the **axis** — stands **perpendicular** to the bases. "Circular" means the bases are circles; "right" means the axis makes a right angle with them, so the solid stands straight up rather than leaning.

This is the difference between a right circular cylinder and the broader idea of a [cylinder](/content/math/geometry/cylinder/index.html): a general cylinder can lean (its axis tilts, making an **oblique** cylinder) or even have non-circular bases. The right circular case is the upright, circular-based one — the form school problems almost always mean, and the one these standard formulas describe...

A right circular cylinder has **two flat circular faces, one curved surface, two circular edges** (the rims), **and no vertices** — no sharp corner anywhere. It belongs to the family of curved solids, alongside the cone and the sphere, rather than the flat-faced [prisms](/content/math/geometry/prism/index.html). You can see how it sits among the others in the guide to [3D geometry shapes](/content/math/geometry/3d-shapes/index.html).

## Volume of a Right Circular Cylinder

The **volume** of a right circular cylinder is:

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

**Where this comes from:** think of the cylinder as a stack of identical circular discs. Each disc has area \(\pi r^2\) — the area of the circular base — and the stack rises to height \(h\). Multiplying the base area by the height gives the total space enclosed: \(\pi r^2 \times h\). This "base area times height" rule is the same one that gives a prism its volume; a cylinder is simply the version with a circular base.

Because the axis is perpendicular to the base, the height \(h\) is also the straight-up distance between the two bases — no extra geometry needed. (In an oblique cylinder you would have to use the perpendicular height, not the slanted length of the side.)

**Variable glossary:** V is the volume, r is the base radius, h is the perpendicular height, and \(\pi \approx 3.14159\). Volume comes out in cubic units (cm³, m³).

## Surface Area of a Right Circular Cylinder

A right circular cylinder has two kinds of surface: the curved side and the two circular ends.

**Curved surface area (CSA)** — the side only:

CSA=\(2\pi rh\)

**Where this comes from:** unroll the curved side and it flattens into a **rectangle**. One pair of sides has length equal to the height \(h\); the other pair has length equal to the circumference of the base, \(2\pi r\) — because the side wraps exactly once around the circular rim. The rectangle's area is height times width: \(h \times 2\pi r = 2\pi rh\).

**Total surface area (TSA)** — the curved side plus both circular bases:

TSA=\(2\pi rh + 2\pi r^2 = 2\pi r(h + r)\)

**Where this comes from:** add the two flat circular ends. Each base is a circle of area \(\pi r^2\), and there are two of them, contributing \(2\pi r^2\). Add that to the curved side \(2\pi rh\) and factor out \(2\pi r\) to get \(2\pi r(h + r)\).

The clearest way to see all three pieces is the cylinder's **net**: the curved side unrolls into a rectangle, with a circle at the top and a circle at the bottom.

| Quantity            | Formula              | Units  |
|--------------------|----------------------|--------|
| Volume             | \(V = \pi r² h\)   | cubic  |
| Curved surface area| \(CSA = 2 \pi r h\)| square |
| Total surface area | \(TSA = 2 \pi r (h + r)\)| square |

## Examples of The Right Circular Cylinder

For consistency, every example below uses centimetres and takes \(\pi \approx \frac{22}{7}\) where it divides cleanly, otherwise \(\pi \approx 3.14\).

### Example 1

**Find the volume of a right circular cylinder with radius 7 cm and height 10 cm. (Use \(\pi \approx \frac{22}{7}\).)**

\[
V=\pi r^2 h \
 = \frac{22}{7} \times 7^2 \times 10 = \frac{22}{7} \times 49 \times 10 = 22 \times 7 \times 10 = 1540 \
\]

**Final answer:** \(V = 1540 \, \text{cm}^3\)

### Example 2

**A right circular cylinder has radius 5 cm and height 8 cm...**

**Final answer:** \(TSA = 408.2 \, \text{cm}^2\)

### Example 3

**Find the curved surface area of a right circular cylinder with radius 4 cm and height 9 cm.**

**Final answer:** \(CSA = 226.08  \, \text{cm}^2\)

### Example 4

**A cylindrical pipe has radius 7 cm and height 20 cm. Find its total surface area.**

**Final answer:** \(TSA = 1188 \, \text{cm}^2\)

### Example 5

**A right circular cylinder has volume 396396396 cm³ and height 9 cm. Find its radius.**

**Final answer:** \(r \approx 3.74 \, \text{cm}\)

### Example 6

**The curved surface of a right circular cylinder is 440 cm² and its radius is 5 cm. Find its height.**

**Final answer:** \(h = 14 \, 	ext{cm}\)

## Conclusion

- A **right circular cylinder** has two equal parallel circular bases and a curved side, with its axis perpendicular to the bases — the upright, circular form.
- It differs from a general or oblique cylinder, which can lean; "right" means the sides stand square to the base.
- **Volume** is \(\pi r^2 h\) — the base area times the height.
- **Curved surface area** is \(2\pi rh\); **total surface area** is \(2\pi r(h+r)\).
- The most common errors are counting one base instead of two and using the diameter as the radius.