Right Circular Cone Calculator: Find A, V, A_L, A_B
Enter any two dimensions of a right circular cone - radius, height, or slant height - and the calculator instantly finds all remaining properties: volume (V), total surface area (A), lateral surface area (A_L), base area (A_B), and the apex angles. You can also solve in reverse from a known volume, lateral area, or total surface area. Switch between metric and imperial units; every step of the math is shown below the result.
Formula
Worked example
A cone with radius r = 5 cm and height h = 12 cm: slant height l = sqrt(5^2 + 12^2) = sqrt(169) = 13 cm. Base area A_B = pi x 5^2 = 78.5398 cm^2. Lateral area A_L = pi x 5 x 13 = 204.2035 cm^2. Total area A = 282.7433 cm^2. Volume V = (1/3) x pi x 25 x 12 = 314.1593 cm^3. Half-angle theta = arctan(5/12) = 22.62 deg.
What is a right circular cone?
A right circular cone is a solid with a flat circular base and a curved lateral surface that tapers to a single point called the apex, with the apex sitting directly above the centre of the base. "Right" refers to the axis being perpendicular to the base, distinguishing it from an oblique cone where the apex is offset to one side. Right circular cones appear throughout engineering and daily life: traffic cones, ice-cream waffle cones, funnels, rocket nose cones, and many roof and tower spires are all approximate right circular cones.
Dimensions, formulas, and what they mean
A right circular cone is fully defined by just two independent measurements. The radius (r) is the distance from the centre of the base circle to its edge. The height (h) is the perpendicular distance from the base centre to the apex. The slant height (l) is the straight-line distance from any point on the base circumference to the apex; by the Pythagorean theorem l = sqrt(r^2 + h^2). If you know any two of r, h, and l, the third follows automatically. The base area A_B = pi x r^2 is just the circle area. The lateral (curved) area A_L = pi x r x l comes from unrolling the cone into a flat sector. Total surface area A = pi x r x (r + l). Volume V = (1/3) x pi x r^2 x h, one third of the cylinder with the same base and height. The half-angle theta = arctan(r / h) at the apex tells you how wide or narrow the cone is; the full aperture angle is 2 x theta.
Reverse-solve modes and when to use them
Most online cone calculators require you to enter radius and height. This calculator also lets you work backwards from a known volume, lateral area, or total surface area paired with the radius, or from slant height paired with either radius or height. That covers common real-world problems: a manufacturer who knows the volume a cone must hold and its base radius can immediately find the required height and surface area for material cost estimation. A designer who knows the overall surface area and base radius can recover the height and volume. Select the appropriate "Solve from" mode before entering values, and the calculator shows every derivation step.
Material efficiency and the optimal cone
If you need to enclose a fixed volume using the least surface area, the optimal right circular cone satisfies the condition r = h x cbrt(2), which gives a height-to-radius ratio of about 1.26:1 and a half-angle of roughly 51.6 degrees. The reference table above lists common aspect ratios and their angles. For aerodynamic applications such as nose cones, half-angles below 15 degrees are typical to minimise drag. For acoustics or optical reflectors, the aperture angle controls how energy is focused or dispersed.
Common right circular cone proportions
| Aspect ratio (h/r) | Shape description | Half-angle (θ) | Typical example |
|---|---|---|---|
| 0.25 | Very flat and wide | ~14.0° | Shallow funnel or party hat |
| 0.5 | Flat cone | ~26.6° | Wide funnel or bowl |
| 1.0 | Equal height and radius | ~45.0° | Balanced general-purpose cone |
| 1.26 | Optimal (min area for V) | ~51.6° | Most material-efficient cone |
| 2.0 | Tall cone | ~63.4° | Canteen or waffle cone |
| 3.0 | Very tall and narrow | ~71.6° | Traffic cone or wizard hat |
How the height-to-radius ratio (aspect ratio) shapes the cone and its angles.
Frequently asked questions
What is the difference between height and slant height?
The height (h) is the straight perpendicular distance from the centre of the base to the apex. The slant height (l) is the distance from any point on the base circumference to the apex, measured along the surface of the cone. Because the base radius adds a horizontal component, l is always greater than h: l = sqrt(r^2 + h^2). When the curved surface is unrolled into a flat shape it becomes a circular sector whose radius equals l.
How do I find the volume if I only know the slant height and radius?
First recover the height: h = sqrt(l^2 - r^2). Then apply V = (1/3) x pi x r^2 x h. For r = 5 cm and l = 13 cm: h = sqrt(169 - 25) = 12 cm, and V = (1/3) x pi x 25 x 12 = 314.16 cm^3. Select "Radius + Slant Height" in the solve mode dropdown and this calculator performs that derivation automatically.
Why is cone volume exactly one-third of the cylinder volume?
This follows from Cavalieri's principle and can be proved rigorously with integral calculus by slicing the cone into thin disks of radius r(y) = r x y/h at height y and integrating pi x r(y)^2 from 0 to h. The result is (1/3) x pi x r^2 x h, exactly one third of the cylinder volume pi x r^2 x h with the same base and height.
What does the half-angle (θ) of a cone mean?
The half-angle theta is the angle between the cone axis (the vertical line from base centre to apex) and the slant side, measured at the apex. It equals arctan(r / h). A small theta means a tall narrow cone; a theta near 90 degrees means an almost flat disk. The aperture angle 2 x theta is the full included angle visible in a cross-section through the axis, and is the dimension usually quoted for conical reflectors, antennas, and nozzles.
How do I find the curved surface area without the slant height?
Compute l = sqrt(r^2 + h^2) first, then A_L = pi x r x l. You can also write this entirely in terms of radius and height: A_L = pi x r x sqrt(r^2 + h^2). For r = 5 cm and h = 12 cm: l = 13 cm and A_L = pi x 5 x 13 = 204.20 cm^2.
Does this calculator work for oblique cones?
No. An oblique cone has its apex offset from the axis, so the slant height varies around the base circle and there is no single formula for the lateral surface area. This calculator applies only to right circular cones, where the apex is directly above the base centre and every generator line (from apex to base edge) has the same length l.