Cone Volume and Surface Area Calculator

Calculate a Cone from Radius and Height or Slant Height

Cone Volume and Surface Area Formula

Formula

\[ V = \frac{1}{3}\pi r^2 h \qquad l = \sqrt{r^2 + h^2} \qquad h = \sqrt{l^2 - r^2} \\ A_L = \pi r l \qquad A_B = \pi r^2 \qquad A = \pi r (r + l) \]

Where:

\(r\) — radius of the circular base
\(h\) — vertical (perpendicular) height from the base to the apex
\(l\) — slant height, measured along the surface from the base edge to the apex
\(V\) — volume
\(A_L\) — lateral (side) surface area
\(A_B\) — base area
\(A\) — total surface area (side + base)

These formulas describe a right circular cone, whose axis is perpendicular to its base. The height and the slant height are linked by the Pythagorean theorem, so knowing either one is enough to find the other.

Cone Volume and Surface Area – Calculation Example

Example: a cone with a base radius of 3 cm and a vertical height of 4 cm (the calculator's default values).

\[ l = \sqrt{r^2 + h^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \text{ cm} \]

\[ V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi \cdot 3^2 \cdot 4 = 12\pi \approx 37.699 \text{ cm}^3 \]

\[ A_L = \pi r l = \pi \cdot 3 \cdot 5 = 15\pi \approx 47.124 \text{ cm}^2 \]

\[ A_B = \pi r^2 = \pi \cdot 3^2 = 9\pi \approx 28.274 \text{ cm}^2 \]

\[ A = A_L + A_B = 15\pi + 9\pi = 24\pi \approx 75.398 \text{ cm}^2 \]

Result: the cone has a volume of about 37.699 cm³, a slant height of 5 cm, a lateral area of 47.124 cm², a base area of 28.274 cm² and a total surface area of 75.398 cm².

How to Use the Calculator

Enter the Radius (r) of the circular base and pick its unit (m, cm, mm, ft or in).
Choose the Second dimension: select Height (h) if you know the vertical height, or Slant height (l) if you know the distance along the side.
Type the value in the field that appears — Height (h) or Slant height (l) — and select its unit. The two units do not have to match; the calculator converts them automatically.
Read the results: Volume, Height (h), Slant height, Lateral (side) surface area, Base area and Total surface area, all expressed in the radius unit, plus a step-by-step calculation trail.

What Is a Right Circular Cone?

A right circular cone is a three-dimensional solid with a circular base and a single apex (tip) that sits directly above the centre of the base. Two different length measurements are commonly used to describe it, and mixing them up is the most frequent source of errors.

The height \(h\) is the straight, perpendicular distance from the centre of the base up to the apex. The slant height \(l\) is the distance from any point on the edge of the base to the apex measured along the sloped surface. The slant height is always the longer of the two (or equal to the height when the radius is zero), and they are related by the Pythagorean theorem: \( l^2 = r^2 + h^2 \). This is why the calculator only needs one of them — the other is derived automatically.

Three area values are reported because they answer different practical questions. The base area tells you how much material the circular bottom covers. The lateral surface area is the area of the curved side — for example, the amount of fabric in a conical tent wall or the label wrapped around a conical container. The total surface area is their sum, which is what you need when a cone is fully covered or painted on all sides, such as an ice-cream cone with a flat top of ice cream.

Interpreting the volume is straightforward: it is the amount of space the cone encloses. A useful fact to remember is that a cone holds exactly one third of the volume of a cylinder with the same base radius and height — the factor \( \tfrac{1}{3} \) in the formula.

Assumptions, Limitations and Tips

The calculator assumes a right circular cone with the apex directly above the centre of the base. Oblique cones (tilted apex) need different formulas for the slant height and are not covered.
If you enter a slant height, it must be at least as large as the radius (\( l \ge r \)). A slant height shorter than the radius describes no real cone, and the calculator will show an error.
Units matter: volume is reported in cubic units (cm³, in³, …) and areas in square units (cm², in², …), always matching the unit you chose for the radius. If the height uses a different unit, it is converted first — you can see this in the calculation steps.
All inputs must be positive. Zero radius or zero height would flatten the cone into a disk or a line, giving a volume of zero.
Results are rounded to three decimal places; \(\pi\) is taken as 3.14159, which is accurate enough for virtually all practical purposes.

A quick sanity check when working by hand: for a fixed radius, the lateral area \( A_L = \pi r l \) grows in direct proportion to the slant height — an example of direct variation. And the square roots in the height–slant height relations are just a special case of taking an nth root with \(n = 2\).

Frequently Asked Questions

How do I calculate the volume of a cone without a calculator?

Multiply the radius by itself, multiply by the height, multiply by \(\pi\), then divide by 3: \( V = \tfrac{1}{3}\pi r^2 h \). For example, a cone with radius 3 cm and height 4 cm gives \( \tfrac{1}{3} \cdot \pi \cdot 9 \cdot 4 = 12\pi \approx 37.7 \) cm³. A quick sanity check: the result should always be less than the volume of a cylinder with the same base and height, since a cone is exactly one third of it.

What is the difference between height and slant height in a cone?

The height \( h \) is the perpendicular distance from the centre of the base straight up to the apex. The slant height \( l \) runs along the sloped surface from the base edge to the apex, so it is always the longer measurement. They are linked by the Pythagorean theorem: \( l^2 = r^2 + h^2 \). If a problem gives you the slant height, first compute \( h = \sqrt{l^2 - r^2} \) before finding the volume.

How do I convert the results between cm, m, ft and in?

The calculator reports lengths, areas and volumes in the unit you chose for the radius. To convert by hand, remember that area and volume conversions involve powers: 1 m = 100 cm means 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³. For imperial units, 1 ft = 12 in, so 1 ft² = 144 in² and 1 ft³ = 1,728 in³. If you enter the height in a different unit, the calculator converts it automatically before calculating.

Why does the calculator say the slant height must be at least as large as the radius?

Because the slant height is the hypotenuse of a right triangle whose legs are the radius and the height. A hypotenuse can never be shorter than either leg, so \( l \ge r \) must hold. If you enter a slant height smaller than the radius, the expression \( h = \sqrt{l^2 - r^2} \) would require the square root of a negative number, which describes no real cone — hence the error message.

How do I find the surface area if only the diameter is known?

Halve the diameter first: \( r = d / 2 \). All cone formulas use the radius, so a diameter of 10 cm means \( r = 5 \) cm. Then apply the usual formulas: lateral area \( A_L = \pi r l \), base area \( A_B = \pi r^2 \), and total surface area \( A = \pi r (r + l) \). Mixing up diameter and radius is one of the most common mistakes and makes results four times too large for areas.

Does the formula work for an oblique (tilted) cone?

Only the volume formula \( V = \tfrac{1}{3}\pi r^2 h \) still works for an oblique circular cone, provided \( h \) is the perpendicular height between the base plane and the apex. The slant height and surface area formulas do not transfer, because the side is no longer symmetric — the distance from the base edge to the apex varies around the circle. This calculator is designed for right circular cones only.

What is a frustum, and how does it relate to a cone?

A frustum is what remains when the top of a cone is sliced off parallel to the base — the shape of a bucket, a lampshade or a truncated tower. Its volume is \( V = \tfrac{1}{3}\pi h (r_1^2 + r_1 r_2 + r_2^2) \), where \( r_1 \) and \( r_2 \) are the two radii. You can also get it by subtracting the volume of the small removed cone from the volume of the full cone.