Skip to main content
Math

Cone Volume & Surface Area Calculator

Enter the base radius r and perpendicular height h of a right circular cone to instantly get the volume V = ⅓ π r² h, slant height l = √(r² + h²), lateral surface area π r l, base area π r², total surface area, base circumference and apex half-angle — useful for ice-cream cones, party hats, traffic cones, funnels, sand piles or any "narrow at the top" geometry problem.

Volume V

314.1593

Lateral area A_l

204.2035

Base area A_b

78.5398

Total surface area A_t

282.7433

Slant height l

13

Diameter d

10

Base circumference C

31.4159

Apex half-angle α

22.6199°

Formula

l = √(r² + h²) ; V = ⅓ π r² h ; A_l = π r l ; A_b = π r² ; A_t = π r (r + l)

Pick any unit you like (cm, m, in, ft). Radius and height share one linear unit; areas use that unit squared; volume uses that unit cubed. h is the perpendicular distance from base to apex — not the slant edge. 1 litre = 1000 cubic centimetres.

Formula

l = √(r² + h²) | V = ⅓ π r² h | lateral A_l = π r l | base A_b = π r² | total A_t = π r (r + l)

Frequently asked

How much ice cream fits in a cone with r = 3 cm and h = 12 cm?

Plug it in: V = ⅓ × π × 3² × 12 = 36π ≈ 113.1 cm³ ≈ 113 ml. In real life ice cream piles above the rim, so a 60–80 ml scoop sits comfortably on top of this cone; if you want the heaped hemisphere too, add V = ⅔ π r³ = 18π ≈ 56.5 cm³.

What happens to volume if I double the radius? Or the height?

Volume scales linearly with height: double h, double V. Radius is squared in the formula (V = ⅓ π r² h), so doubling r quadruples the volume (2² = 4). A cone with r×2 holds 4 of the original; with h×2 it holds only 2. And a cone is always exactly 1⁄3 the volume of the cylinder with the same base and height (Euclid).

I only know the slant height l and the diameter — what now?

Halve the diameter to get the radius r, then recover the perpendicular height with h = √(l² − r²) and feed both into the tool. Example party hat: l = 25 cm, diameter 16 cm (r = 8 cm), so h = √(625 − 64) = √561 ≈ 23.7 cm. The tool then gives V = ⅓ π × 64 × 23.7 ≈ 1587 cm³ and lateral area π × 8 × 25 = 200π ≈ 628 cm² — the paper area needed for the cone pattern.

Why does the total surface area include the base? My ice-cream cone doesn't have one.

A closed solid cone (a traffic cone, a sand-pile model) has a circular base, so A_t = lateral + base = π r (l + r). For an open cone (ice-cream cone, party hat, Christmas-tree ornament) just use the lateral area A_l = π r l. The tool reports both — pick whichever matches the shape you are building.

Related tools

Percentage Calculator

Percent of, percent change, and percent add/subtract in one.

GCD & LCM Calculator

Enter 2–6 positive integers to get the greatest common divisor (HCF / GCD) and least common multiple (LCM), with the Euclidean step chain shown.

Average Calculator (Mean / Median / Mode)

Enter a list of numbers to get the mean, median, mode, range plus standard deviation, variance and total.

Quadratic Equation Solver

Enter the coefficients of ax² + bx + c = 0 to find the real or complex roots, discriminant and vertex.

Password Strength (Entropy) Calculator

Estimate a password's bit entropy, brute-force time and strength tier. All computation happens in your browser.

Scientific Notation Converter

Convert between standard decimal numbers and scientific notation, with significant figures and order of magnitude.

Permutations & Combinations (nPr / nCr) Calculator

Compute permutations P(n,r), combinations C(n,r) and factorial n! — useful for probability problems, lottery odds and combinatorics homework.

Standard Deviation Calculator

Paste a list of numbers to compute mean, median, sample and population variance and standard deviation — with the working shown.

Triangle Calculator (SSS / SAS / ASA)

Solve a triangle from 3 sides, 2 sides + 1 angle, or 2 angles + 1 side — area, perimeter and remaining parts via the law of sines / cosines.

Pythagorean Theorem Calculator

Given any two sides of a right triangle (two legs, or one leg plus the hypotenuse), instantly find the third side, area, perimeter and the two non-right angles.

Circle Calculator (radius / diameter / circumference / area)

Enter any one of radius, diameter, circumference or area to get the other three — useful for design, engineering and DIY.

Roman Numeral Converter

Two-way conversion between Arabic numbers (1–3999) and Roman numerals (I, V, X, L, C, D, M) — handy for typesetting, chapter titles and homework.

Slope & Line Equation Calculator (y = mx + b from Two Points)

Enter two points (x₁, y₁) and (x₂, y₂) to instantly get the slope, y-intercept, line equation y = mx + b, distance, and midpoint — a classroom staple for algebra and coordinate geometry.

Birthday Paradox Calculator

Enter group size n to see the probability that at least two people share a birthday — the classic birthday problem.

Logarithm Calculator (log / ln / log₂ / any base)

Compute logₐ(x) for any base — natural log (ln), common log (log₁₀), binary log (log₂) and a custom base, with the change-of-base steps shown.

Z-Score (Standard Score) Calculator

Enter a value, the mean and the standard deviation to compute the z-score and the corresponding normal-distribution percentile and probabilities.

Screen Pixel Density (PPI) Calculator

Enter the screen resolution and diagonal size to get pixel density (PPI), real width/height, dot pitch and total pixel count.

Hong Kong Mark Six Odds Calculator

Enter the number of tickets / selections and see the actual probability of hitting first, second … prizes in a Mark Six (6-of-49) draw.

Decimal to Fraction Converter

Convert any decimal (including repeating decimals) to a simplified fraction and a mixed number.

Sphere Volume & Surface Area Calculator

Give a sphere any one of radius, diameter, surface area or volume and instantly get the other three — plus the great-circle circumference and area.

Cylinder Volume & Surface Area Calculator

Enter the radius and height of a cylinder to get volume (π r²h), lateral surface, base area and total surface area.

Permutations (nPr) Calculator

Compute nPr — the number of ordered arrangements of r items chosen from n — alongside n!, r! and the related combination nCr.

Prime Factorization Calculator

Factor any integer from 2 up to 10¹² into primes, see the canonical exponent form, count and list all divisors.

Geometric Mean Calculator

Compute the n-th root of the product of n positive numbers — the right average for growth rates, returns and ratios — alongside the arithmetic mean for comparison.

Fibonacci Sequence Calculator

Enter any integer n from 0 to 1500 to instantly compute F(n) and F(n−1) with BigInt precision, the consecutive-term ratio (converging to the golden ratio φ) and the first 30 terms of the sequence.

Dice Roll Probability Calculator

Pick the number of dice, sides (d4 / d6 / d8 / d10 / d12 / d20) and a target sum to compute the probability of rolling exactly that total, at least, or at most.

Arithmetic Series Calculator

Enter first term a, common difference d and number of terms n to compute the nth term aₙ and the partial sum Sₙ = n/2·(2a + (n − 1)d).

Survey Sample Size Calculator

Enter confidence level, margin of error, expected proportion (and optional population size) to compute the survey sample size you need.

Geometric Series Sum Calculator

Enter the first term a, common ratio r and number of terms n to find the sum of the first n terms of a geometric series — plus the infinite-series sum when |r| < 1.