# Right Circular Cone: Volume, Surface Area, and Examples

## TL;DR

A right circular cone is a 3D solid with one circular base and a single apex that sits directly above the centre of that base. Its volume is \( V = \frac{1}{3}\pi r^2 h \), its curved surface area is \( CSA = \pi r l \), and its total surface area is \( TSA = \pi r(l + r) \), where \( l \) is the slant height.

## What is a Right Circular Cone?

A **right circular cone** is a three-dimensional solid with a **flat circular base** and a curved surface that tapers smoothly to a single point called the **apex** (or vertex), where the apex sits **directly above the centre of the base**. "Circular" means the base is a circle; "right" means the line from the apex to the base centre — the **axis** — meets the base at a right angle.

- The **radius (r)** is the radius of the circular base.
- The **height (h)** is the perpendicular distance straight up from the base centre to the apex.
- The **slant height (l)** is the distance from any point on the rim, straight up the slanted surface, to the apex.

A right circular cone has **two surfaces** (one flat circular base, one curved lateral surface), **one curved edge** (the rim), **and one vertex** (the apex).

## Height, Slant Height, And The Right Triangle That Links Them

The **height (h)** goes straight up the middle. The **slant height (l)** runs along the outside surface. Because the cone is _right_, the radius \( r \), the height \( h \), and the slant height \( l \) form a **right triangle** inside the cone, with \( l \) as the hypotenuse. By the [Pythagorean theorem](/content/math/geometry/pythagoras-theorem/index.html):

\[ l = \sqrt{r^2 + h^2} \]

## Volume of A Right Circular Cone

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

\[ V = \frac{1}{3}\pi r^2 h \]

## Surface Area of a Right Circular Cone

**Curved (lateral) surface area (CSA)**:

\[ CSA = \pi r l \]

**Total surface area (TSA)**:

\[ TSA = \pi r(l + r) \]

| Quantity               | Formula                 | Units      |
|------------------------|-------------------------|------------|
| Slant height           | \( l = \sqrt{r^2 + h^2} \) | length     |
| Volume                 | \( V = \frac{1}{3}\pi r^2 h \) | cubic      |
| Curved surface area    | \( CSA = \pi r l \)   | square     |
| Total surface area     | \( TSA = \pi r (l + r) \) | square     |

## Examples of the Right Circular Cone

For consistency, every example below uses centimetres and takes \( \pi \approx 3.14 \).

### Example 1

**A right circular cone has radius 3 cm and height 4 cm. Find its slant height.**

\[ l = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm} \]

### Example 2

**A right circular cone has radius 7 cm and height 9 cm. A student finds the total surface area using the height instead of the slant height. Find the correct total surface area.**

Correct slant height:
\[ l = \sqrt{7^2 + 9^2} = \sqrt{49 + 81} = \sqrt{130} \approx 11.4 \text{ cm} \]

\[ TSA = 3.14 \times 7 \times (11.4 + 7) = 404.4 \text{ cm}^2 \]

### Example 3

**Find the volume of a right circular cone with radius 6 cm and height 10 cm.**

\[ V = \frac{1}{3} \times 3.14 \times (6^2) \times 10 \approx 376.8 \text{ cm}^3 \]

### Example 4

**Find the curved surface area of a right circular cone with radius 5 cm and slant height 13 cm.**

\[ CSA = 3.14 \times 5 \times 13 \approx 204.1 \text{ cm}^2 \]

### Example 5

**A right circular cone has radius 8 cm and slant height 17 cm. Find its total surface area.**

\[ TSA = 3.14 \times 8 \times (17 + 8) = 628 \text{ cm}^2 \]

### Example 6

**A right circular cone has volume 100π cm³ and radius 5 cm. Find its height.**

\[ h = \frac{100 \times 3}{25} = 12 \text{ cm} \]

## Conclusion

- A **right circular cone** has a circular base and an apex directly above the base centre, so its axis is perpendicular to the base.
- **Volume** is \( \frac{1}{3}\pi r^2 h \) — exactly one-third of the matching cylinder.
- **Curved surface area** is \( \pi r l \); **total surface area** is \( \pi r(l + r) \).