Cone: Definition, Volume, Surface Area Formulas, and Examples
Cone: Definition, Volume, Surface Area Formulas, and Examples
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 and a triangular pyramid 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 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:
[ 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.