CalConverters

Cone Calculator: Volume, Slant Height & Surface Area

Find a cone's volume, slant height, lateral area, and total surface area from its radius plus height, slant, or volume — with exact step-by-step results.

Loading calculator…

The cone's measures

A right circular cone is defined by its base radius rr and height hh — a circular base tapering straight up to a single apex directly above the center. From those two, every other measure follows, starting with the slant height (the hypotenuse of the r/h right triangle, via the Pythagorean theorem):

s=r2+h2s = \sqrt{r^2 + h^2} V=13πr2hL=πrsB=πr2A=πr(s+r)V = \frac{1}{3}\pi r^2 h \qquad L = \pi r s \qquad B = \pi r^2 \qquad A = \pi r(s + r)

Volume is exactly one-third of a cylinder with the same base and height — a classic calculus result. Lateral area comes from unrolling the cone's curved side flat into a circular sector of radius ss.

Tip: The most common slip is forgetting the 13\frac{1}{3} in the volume formula, or using the height hh instead of the slant ss in the lateral-area formula (L=πrsL = \pi rs, never πrh\pi rh).

Worked example: r = 3, h = 4

s=9+16=25=5V=13π(9)(4)=12π≈37.6991s = \sqrt{9 + 16} = \sqrt{25} = 5 \qquad V = \frac{1}{3}\pi(9)(4) = 12\pi \approx 37.6991 L=π(3)(5)=15π≈47.1239B=π(9)≈28.2743A=π(3)(8)=24π≈75.3982L = \pi(3)(5) = 15\pi \approx 47.1239 \qquad B = \pi(9) \approx 28.2743 \qquad A = \pi(3)(8) = 24\pi \approx 75.3982

This is the classic 3-4-5 right triangle in disguise, so the slant height comes out to an EXACT whole number (5), and every π\pi-based measure has a clean exact symbolic form too. The half apex angle is θ=arctan⁡(3/4)≈36.87°\theta = \arctan(3/4) \approx 36.87° and the base angle is β=90°−36.87°≈53.13°\beta = 90° - 36.87° \approx 53.13°.

Worked example: r = 6, h = 8

s=36+64=100=10V=13π(36)(8)=96π≈301.5929s = \sqrt{36 + 64} = \sqrt{100} = 10 \qquad V = \frac{1}{3}\pi(36)(8) = 96\pi \approx 301.5929 L=π(6)(10)=60π≈188.4956A=π(6)(16)=96π≈301.5929L = \pi(6)(10) = 60\pi \approx 188.4956 \qquad A = \pi(6)(16) = 96\pi \approx 301.5929

Doubling every side of the 3-4-5 triangle to 6-8-10 keeps the slant exact. Volume and total surface area happen to land on the same number here (96π96\pi) — a coincidence of these particular values, not a general rule.

Solving in the other direction

Tip: The calculator accepts radius with height, slant height, OR volume — pick whichever two measures you already know.
KnownSolve for height
rr, ssh=s2−r2h = \sqrt{s^2 - r^2} (requires s>rs > r)
rr, VVh=3V/(πr2)h = 3V/(\pi r^2)

Given radius 3 and slant height 5, h=25−9=16=4h = \sqrt{25 - 9} = \sqrt{16} = 4 — matching the worked example above exactly, since 33-44-55 is a right triangle.

Common mistakes

  • Forgetting the 13\frac{1}{3} in the volume formula. A cone holds exactly one-third of the cylinder with the same base and height — omitting it triples the true volume.
  • Confusing height with slant height. The height hh is vertical (apex to base center); the slant ss runs along the surface (apex to rim), and ss is always the longer of the two.
  • Using hh instead of ss in the lateral-area formula. Lateral area is L=πrsL = \pi r s, never πrh\pi r h — the curved surface unrolls along the slant, not the height.

Where this shows up

  • Ice cream cones, funnels, piles of sand or gravel: the volume formula estimates capacity or quantity for anything cone-shaped.
  • Party hats, traffic cones, roof spires: the lateral area tells you how much material (paper, plastic, fabric) covers the curved surface.
  • Building block for other solids: slicing off a cone's tip gives a conical frustum, and a cone often caps a cylinder or sphere in composite shapes.

A cone is fully determined by any two of radius, height, slant height, and volume — this calculator solves for the missing pair first via the Pythagorean relationship or the volume formula, then derives the slant height, volume, lateral area, base area, total surface area, and both angles from that resolved radius-and-height pair.

Frequently asked questions

What are the formulas for a cone?
With radius r, height h, and slant height s = √(r² + h²): volume V = (1/3)πr²h, lateral (side) area L = πrs, base area B = πr², and total surface area A = πr(s + r) = L + B.
Why does a cone's volume use 1/3 and a cylinder's doesn't?
A cone with the same base and height as a cylinder holds exactly one-third the volume — a classic result from calculus (or, physically, you can pour three cone-fuls of sand into a same-size cylinder to fill it). Forgetting the 1/3 is the single most common cone mistake.
What is slant height, and how is it different from height?
Height h is the straight vertical distance from the apex down to the center of the base. Slant height s is the distance from the apex down along the cone's outer surface to the rim — the hypotenuse of the right triangle formed by r and h, so s = √(r² + h²) and s is always longer than both r and h.
How do I find the height if I only know the radius and slant height?
Rearrange s = √(r² + h²) for h: h = √(s² − r²). For example, radius 3 and slant height 5 gives h = √(25 − 9) = √16 = 4 — the classic 3-4-5 right triangle. The slant height must be greater than the radius, or no real cone exists.
How do I find the height if I only know the radius and volume?
Rearrange V = (1/3)πr²h for h: h = 3V/(πr²). For example, radius 3 and volume 37.6991 gives h = 3(37.6991)/(π × 9) ≈ 4.
What is the most common mistake with cone surface area?
Using the height h instead of the slant height s in the lateral area formula. Lateral area is always L = πrs (the slant, unrolled into a flat sector), never πrh — mixing up the two systematically understates the surface area.
What are the angles of a cone?
The half apex angle θ = atan(r/h) is the angle between the cone's axis and its slant surface; the base angle β = 90° − θ is the angle the slant makes with the base. A tall, narrow cone has a small θ; a short, wide cone has a large θ.
How is the cone related to the cylinder and the sphere?
A cone's volume is exactly one-third of a cylinder sharing the same base radius and height — both are built on the same πr² base term. Slicing off a cone's tip parallel to the base leaves a conical frustum, and a cone often caps a hemisphere (half a sphere) in composite-solid problems.

Related calculators