# Cone: Definition, Volume, Surface Area Formulas, and Examples

[Geometry](/content/tag/geometry/index.html)

### TL;DR

A cone is a 3D solid with one circular base that narrows to a single point called the apex. Its volume is \( \frac{1}{3}\pi r^2 h \) — exactly one-third of a cylinder with the same base and height — and its total surface area is \( \pi r(l + r) \), where l is the slant height. This article derives both formulas, explains the difference between height and slant height, and shows the worked examples and slips that trip students up.

### What Is A Cone?

A **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). The straight-line distance from the center of the base to the apex, measured perpendicular to the base, is the **height (h)**. The distance from any point on the edge of the base straight up the slanted surface to the apex is the **slant height (l)**.

When the apex sits directly above the center of the base, it is a **right circular cone** — the standard case and the one these formulas describe. A cone is a close cousin of the cylinder: a cylinder has two equal circular ends, while a cone collapses one of them to a point. It also sits alongside the pyramids — a [rectangular pyramid](/content/math/geometry/rectangular-pyramid/index.html) and a [triangular pyramid](/content/math/geometry/triangular-pyramid/index.html) do the same "base narrowing to a point" trick over a flat-sided base instead of a circle. Unlike a prism, which keeps a constant cross-section, a cone tapers. All belong to the family of [geometric shapes](/content/math/geometry/geometric-shapes/index.html) you meet in solid geometry.

### 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. 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):

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

The slant height is always longer than the height, because it is the hypotenuse of that triangle. If a problem gives you h and r, you compute l from this before touching the surface-area formula.

### Volume Of A Cone

The **volume** of a cone is:

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

**Where this comes from:** \( \pi r^2 \) is the area of the circular base, and \( \pi r^2 h \) would be the volume of a _cylinder_ with that base and height. A cone fills exactly one-third of that cylinder — the water-pouring fact from the top of this article — so multiply by \( \frac{1}{3} \). The one-third is the same factor that turns any prism into the matching pyramid; it is the signature of a solid that tapers to a point.

**Variable glossary:** V is 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 Cone

A cone has two surfaces: the curved side and the flat circular bottom.

**Curved (lateral) surface area** — the slanted part only:

\[ 
CSA = \pi r l 
\]

**Total surface area** — the curved part plus the circular base:

\[ 
TSA = \pi r l + \pi r^2 = \pi r (l + r)  
\]

**Where the curved surface area comes from:** unroll the slanted surface of a cone and it flattens into a sector (a slice) of a large circle whose radius is the slant height l. Working out the area of that sector gives exactly \( \pi r l \). This is the "net" of a cone, and it is the clearest way to see why l — not h — drives the surface area.

**Variable glossary:** CSA is the curved surface area, TSA is the total surface area, l is the slant height, r is the base radius. Surface area comes out in square units (cm², m²).

| Quantity               | Formula                        | Units   |
|-----------------------|--------------------------------|---------|
| Slant height          | \( l = \sqrt{r² + h²} \)    | length  |
| Volume                | \( V = \frac{1}{3} \pi r² h \) | cubic   |
| Curved surface area   | \( CSA = \pi r l \)        | square  |
| Total surface area    | \( TSA = \pi r (l + r) \)  | square  |

### Examples Of The Cone

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

#### Example 1

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

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

\[ l = \sqrt{3^2 + 4^2} \]

\[ l = \sqrt{9 + 16} \]

\[ l = \sqrt{25} \]

**Final answer:** l = 5 cm

#### Example 2

**A 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.**

Take the wrong path first, because using h in place of l is the classic cone error.

Wrong attempt: the student writes TSA = \( \pi r(h + r) \) and plugs in h = 9.

\[ TSA = 3.14 \times 7 \times (9 + 7) = 3.14 \times 7 \times 16 = 351.7 \text{ cm²} \]

The break: the slant surface follows the _slanted_ edge, not the vertical height. The formula needs l, and l is longer than h, so this answer is too small.

Correct method: first find the slant height.

\[ l = \sqrt{r^2 + h^2} = \sqrt{7^2 + 9^2} = \sqrt{49 + 81} = \sqrt{130} \approx 11.4 \text{ cm} \]

Now use it.

\[ TSA = \pi r(l + r) = 3.14 \times 7 \times (11.4 + 7) = 3.14 \times 7 \times 18.4 \]

**Final answer:** \( \approx 404.4 \text{ cm²} \)

#### Example 3

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

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

\[ V = \frac{1}{3} \times 3.14 \times 6^2 \times 10 \]

\[ V = \frac{1}{3} \times 3.14 \times 36 \times 10 \]

\[ V = \frac{1}{3} \times 1130.4 \]

**Final answer:** \( \approx 376.8 \text{ cm³} \)

#### Example 4

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

\[ CSA = \pi r l \]

\[ CSA = 3.14 \times 5 \times 13 \]

\[ CSA = 3.14 \times 65 \]

**Final answer:** \( \approx 204.1 \text{ cm²} \)

#### Example 5

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

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

\[ TSA = 3.14 \times 8 \times (17 + 8) \]

\[ TSA = 3.14 \times 8 \times 25 \]

\[ TSA = 3.14 \times 200 \]

**Final answer:** \( \approx 628 \text{ cm²} \)

#### Example 6

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

Start from the volume formula and solve for h.

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

\[ 100	ext{π} = \frac{1}{3} \times \pi \times 5^2 \times h \]

\[ 100	ext{π} = \frac{1}{3} \times \pi \times 25 \times h \]

Divide both sides by π.

\[ 100 = \frac{1}{3} \times 25 \times h \]

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

\[ h = 100 \times 3 \div 25 \]

\[ h = 300 \div 25 \]

**Final answer:** h = 12 cm

### Why One-Third Matters Beyond The Classroom

The factor of one-third looks like a small detail until you are pouring concrete.

Anyone who fills a conical mould — a pile of grain, a heap of sand, a concrete pour that tapers — needs the one-third or they badly over- or under-estimate the material. A conical pile of gravel that _looks_ about as big as a cylindrical bin holds only a third as much.

Architects sizing a spire, manufacturers casting a funnel, and engineers modelling a stockpile all lean on \( \frac{1}{3}\pi r^2 h \). The shape shows up wherever something must channel down to a point: a funnel directing liquid, a loudspeaker spreading sound, a rocket nose-cone cutting air. In each case the geometry — circular base, single apex, that fixed one-third — is doing real work.

### Tripping Points To Avoid

#### Mistake 1: Using height instead of slant height in surface area

**Where it slips in:** any surface-area calculation when the problem gives height, not slant height.

**Don't do this:** plug h straight into \( \pi r(l + r) \). The lateral surface follows the slanted edge.

**The correct way:** compute \( l = \sqrt{r^2 + h^2} \) first, then use l. The slant height is always the larger of the two. The rusher who skips this step gets a surface area that is reliably too small.

#### Mistake 2: Forgetting the one-third in volume

**Where it slips in:** the volume formula, especially right after studying cylinders.

**Don't do this:** write \( V = \pi r^2 h \). That is the cylinder's volume, three times too big for a cone.

**The correct way:** a cone is one-third of its enclosing cylinder, so \( V = \frac{1}{3}\pi r^2 h \). The memorizer who carries the cylinder formula over forgets the shape tapers to a point.

#### Mistake 3: Mixing up CSA and TSA

**Where it slips in:** questions that ask for "surface area" without saying which.

**Don't do this:** report \( \pi r l \) when the question wants the whole closed solid, or add the base when only the curved part is wanted (for example, an open funnel or a party hat has no base).

**The correct way:** read whether the base is included. \( CSA = \pi r l \) is the slanted side only; \( TSA = \pi r(l + r) \) adds the circular base. A closed cone gets TSA; an open one gets CSA.

### Conclusion

- A **cone** has one circular base narrowing to an apex; the standard case is the right circular cone.

- **Slant height** \( l = \sqrt{r^2 + h^2} \) is always longer than the height h and drives the surface area.

- **Volume** is \( V = \frac{1}{3}\pi r^2 h \) — exactly one-third of the matching cylinder.

- **Curved surface area** is \( 
CSA = \pi r l \); **total surface area** is \( TSA = \pi r (l + r) \), adding the circular base.

- The most common error is using h where the formula needs l.

### A Practical Next Step

Test your understanding with these problems: take any cone given by radius and height, find its slant height first, then compute volume and total surface area in that order. If you confuse which surface-area formula to use, return to the unfolding net above and ask whether the base is part of the solid.