Cone Calculator

Math Geometry • Three Dimensional Geometry

Written by STEM Calculators Team Published February 1, 2026 Updated February 24, 2026

Cone Volume & Surface Area Calculator – Right or Oblique (Inclined)

Compute volume \(V=\tfrac13\pi r^2 h\), lateral area, and total surface area for right cones (with \(l=\sqrt{r^2+h^2}\)) and oblique cones using \(A_L \approx \pi r\,l_{\text{avg}}\).

Inputs accept 1e-3, pi, e, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Cone size

Given: Value: Perpendicular height \(h\): Units label (optional):

Right vs oblique (inclined)

Type:

Slant height \(l\) (optional override):

If you enter \(l\), the tool uses it for lateral area \(A_L=\pi r l\) and reports how it compares to \(\sqrt{r^2+h^2}\).

Animation progress:

Drag to orbit • Shift+drag to pan • wheel/trackpad to zoom. Double-click the view to fit.

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Projection: Wireframe density: Show axes:

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3D cone view (interactive)

The slider animates the tilt by moving the apex sideways. Labels point to the exact dimensions they represent.

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Theory

Right cones: what the measurements mean

A cone is a solid with a circular base and a single apex (tip). Every point on the rim of the base is connected to the apex by a straight segment called a generator. The most common case is a right circular cone, where the apex sits directly above the center of the base. In that right case, three lengths are especially important:

Radius \(r\): distance from the base center to the rim.
Perpendicular height \(h\): shortest distance from apex to the base plane.
Slant height \(l\): length of a generator from apex to the rim.

For a right cone, \(r\), \(h\), and \(l\) form a right triangle through the base center and a rim point, so: \[ \begin{aligned} l=\sqrt{r^2+h^2}. \end{aligned} \] This relationship is why many formulas involve \(l\): it measures the “tilted” side length that creates the lateral surface.

Volume of a cone

The cone volume is one third of the volume of a cylinder with the same base and height. The base area is \(A_b=\pi r^2\), and the volume is: \[ \begin{aligned} V=\tfrac13 A_b h=\tfrac13 \pi r^2 h. \end{aligned} \] The key point is that \(h\) is the perpendicular height (distance to the base plane), not the slant height. This makes the volume formula stable even when the cone is tilted: as long as the base circle is unchanged and the perpendicular height is the same, the volume is governed by \(\tfrac13 \pi r^2 h\).

Surface area: lateral vs total

The curved side of the cone is the lateral surface. For a right cone, if you cut the lateral surface along one generator and flatten it, you obtain a circular sector. The sector’s radius is the slant height \(l\), and its arc length equals the circumference of the base \(2\pi r\). From that geometry, the lateral area becomes: \[ \begin{aligned} A_L=\pi r l. \end{aligned} \] The total surface area includes the base disk: \[ \begin{aligned} A_T=A_L + A_b=\pi r l + \pi r^2. \end{aligned} \] In applications, \(A_L\) is what you paint or wrap on the side, while \(A_T\) is what you would cover if the base is included (for example, a closed container).

Oblique (inclined) cones and the \(l_{\text{avg}}\) approximation

An oblique cone is still formed from a circular base and an apex, but the apex is not directly above the base center. As a result, the generator lengths vary around the rim: generators on the “near” side are shorter and those on the “far” side are longer. That variation makes the exact lateral area more complicated than the right-cone sector argument.

A common practical approximation is to replace the varying generators by an average generator length \(l_{\text{avg}}\), then use: \[ \begin{aligned} A_L \approx \pi r\,l_{\text{avg}}. \end{aligned} \] This calculator supports two ways to get \(l_{\text{avg}}\): (1) you can enter it directly, or (2) you can define the tilt using a horizontal offset \(d\) of the apex’s projection on the base plane, and the tool numerically averages the generator lengths around the rim. The volume still uses the perpendicular height \(h\): tilting changes the side shape, but the “one third of the prism/cylinder” relationship with the same base and height remains tied to \(h\), not to the slanted edges.

Tip: If you care about packaging/material cost, surface area is the driver (and tilt can matter). If you care about capacity, volume is the driver (and it depends on \(r\) and perpendicular \(h\)).

Frequently Asked Questions

What formulas are used for a right cone?

Right cone formulas are: V = (1/3)πr^2h, l = √(r^2 + h^2), A_L = πrl, and A_T = πr(r + l).

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

Height h is perpendicular to the base plane and is used in the volume formula. Slant height l lies along the cone’s lateral surface from the rim to the apex and is used in surface area formulas for right cones.

Does an oblique cone have the same volume as a right cone?

Volume depends on base area and perpendicular height: V = (1/3)πr^2h_perp. If an oblique cone has the same base radius and the same perpendicular height as a right cone, their volumes match.

Why is the lateral area of an oblique cone only approximate here?

For an oblique cone, generator lengths vary around the rim, so there is no single slant height like the right cone case. An average generator length l_avg provides a practical approximation for comparison, but it may not match an exact integral-based surface area in all geometries.

What does the sample input produce?

Example: r = 4, h = 9 (right cone) gives V ≈ (1/3)π·16·9 ≈ 150.8 and l = √(16 + 81) = √97 ≈ 9.8489, so A_L ≈ π·4·l ≈ 123.8 and A_T ≈ π·4(4 + l) ≈ 174.1. If an inclined case uses l = 10 as a representative generator, an approximate lateral area is A_L ≈ π·4·10 ≈ 125.7.

Cone Volume & Surface Area Calculator – Right or Oblique (Inclined)

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Frequently Asked Questions

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