VolLab

Cone Volume Calculator

A third of the cylinder that contains it.

Radius

Height

Volume

150.796

Surface area: 174.03
Slant height: 9.849

What you entered

Radius: 4Height: 9

1
Volume = ⅓ × π × r² × h
(1/3) × π × 4² × 9= 150.7964
2
Slant height
√(4² + 9²)= 9.8489
3
Total surface area = πr(r + slant)
π × 4 × (4 + 9.8489)= 174.0299

Result

Volume: 150.796

A cone with radius 4 and height 9 has a volume of 150.796 and a total surface area of 174.03.

Cone volume, surface area and slant height

Volume is one third pi r squared h - exactly a third of a cylinder with the same base and height. The slant height comes from Pythagoras on the radius and height.

Radius x Height	Volume	Total surface area	Slant height
3 x 4	37.70	75.40	5.000
4 x 6	100.53	140.88	7.211
5 x 12	314.16	282.74	13.000
6 x 8	301.59	301.59	10.000
8 x 15	1005.31	628.32	17.000
10 x 20	2094.40	1016.64	22.361

The rows with whole-number slant heights - 3 x 4, 5 x 12, 6 x 8, 8 x 15 - are the Pythagorean triples again, since the radius, height and slant height form a right triangle. The one third factor is the thing to remember: a cone holds exactly a third of the cylinder that would enclose it, which is why an ice cream cone holds so much less than its silhouette suggests. Surface area here includes the circular base as well as the curved side.

Why the ⅓

A cone's volume is always exactly one-third of the cylinder that shares its base and height — a relationship you can verify by filling a cone-shaped cup with water and pouring it into a same-sized cylinder three times.

Slant height vs. height

The height is the straight perpendicular line from apex to base center; the slant height (used for surface area) runs along the cone's outer surface, and is always longer — found via the Pythagorean theorem.

Frequently asked questions

How do I calculate the volume of a cone?

Volume = ⅓ × π × r² × h. A cone with radius 4 cm and height 9 cm: V = ⅓ × π × 16 × 9 = 150.8 cm³. It's exactly one-third of a cylinder with the same base and height.

What is slant height?

Slant height (l) is the distance from the tip of the cone to any point on the edge of the base, measured along the surface. l = √(r² + h²). For radius 3 and height 4: l = √(9+16) = 5. It's always longer than the vertical height.

How do I find the surface area of a cone?

Total surface area = πr² + πrl, where l is the slant height. The first term is the circular base; the second is the lateral (curved) surface. For r=3, h=4, l=5: SA = π×9 + π×3×5 = 9π + 15π = 75.4.