314.1593
cu units
13
78.5398
sq units
204.2035
sq units
282.7433
sq units
10
2.4
45.2397
deg
314.1593
cu units
13
78.5398
sq units
204.2035
sq units
282.7433
sq units
10
2.4
45.2397
deg
The Cone Calculator computes the volume, slant height, total surface area, lateral surface area, and base area of a right circular cone. A cone is a three-dimensional solid with a circular base that tapers smoothly to a single point called the apex or vertex.
The key formulas for a right circular cone are:
$$V = \frac{1}{3}\pi r^2 h$$
$$l = \sqrt{r^2 + h^2}$$
$$SA_{\text{total}} = \pi r(r + l)$$
$$SA_{\text{lateral}} = \pi r l$$
where $$r$$ is the base radius, $$h$$ is the perpendicular height, and $$l$$ is the slant height. The volume formula shows that a cone's volume is exactly one-third of the volume of a cylinder with the same base and height -- a remarkable result first proven by Eudoxus of Cnidus and later formalized by Archimedes.
The slant height $$l = \sqrt{r^2 + h^2}$$ follows directly from the Pythagorean theorem applied to the right triangle formed by the radius, height, and slant height of the cone. This is the distance measured along the surface from the base edge to the apex.
Cones appear throughout engineering and manufacturing. Conical hoppers and funnels use the volume formula for capacity calculations. The angle of the cone walls determines the flow characteristics of granular materials -- too steep and material bridges, too shallow and flow stagnates. Conical roof structures, rocket nose cones, and traffic cones all rely on cone geometry for their design.
In materials and packaging, the lateral surface area determines how much material is needed to construct a cone. When you "unroll" a cone's lateral surface, it forms a circular sector (a fan shape) whose arc length equals the base circumference $$2\pi r$$ and whose radius equals the slant height $$l$$. This unrolled pattern is the basis for constructing conical paper cups, party hats, and sheet metal funnels.
Optics and acoustics rely on conical geometry. Conical reflectors focus parallel light or sound waves to a point. Speaker cones vibrate to produce sound, and their surface area affects radiation efficiency. The conical horn antenna in telecommunications uses the flaring angle (related to $$r/h$$ ratio) to control beam width and gain.
In geology and volcanology, volcanic cones and cinder cones approximate conical shapes. Geologists use cone volume formulas to estimate the amount of material ejected during eruptions. Sand piles and grain piles naturally form cones at the material's angle of repose, and their volume is calculated using this formula.
This comprehensive calculator provides all five key cone measurements from just the radius and height, making it a complete tool for any conical geometry problem.
Enter the base radius and height. The calculator first computes the slant height using the Pythagorean theorem: $$l = \sqrt{r^2 + h^2}$$. Then it applies the standard cone formulas: volume $$= \frac{1}{3}\pi r^2 h$$, lateral SA $$= \pi r l$$, total SA $$= \pi r(r + l)$$, and base area $$= \pi r^2$$.
The Volume is the space inside the cone. The Slant Height is the distance from the base edge to the apex along the surface. The Total Surface Area includes the curved side and the circular base. The Lateral Surface Area is the curved side only (useful for material-cutting patterns). The Base Area is the flat circular bottom. Use consistent units for radius and height.
Inputs
Results
V = (1/3)*pi*25*12 = 100*pi ~ 314.16. Slant = sqrt(25+144) = sqrt(169) = 13. Lateral SA = pi*5*13 = 65*pi ~ 204.20.
Inputs
Results
V = (1/3)*pi*100*3 = 100*pi ~ 314.16. Note: same volume as the previous example but very different proportions. Slant = sqrt(100+9) ~ 10.44.
This can be proven using integral calculus. Stacking infinitesimal circular disks from base to apex, each disk at height $$x$$ has radius $$r(1-x/h)$$. Integrating $$\pi r^2(1-x/h)^2$$ from 0 to $$h$$ yields $$\frac{1}{3}\pi r^2 h$$. Alternatively, Cavalieri's principle shows three identical pyramids fill a cube, giving the 1/3 factor for any cone or pyramid.
The slant height $$l = \sqrt{r^2 + h^2}$$ is the straight-line distance from the base edge to the apex along the cone surface. It is essential for calculating lateral surface area and for constructing the unrolled pattern of a cone from flat material. Without the slant height, you cannot determine material requirements for cone fabrication.
The half-apex angle $$\alpha$$ (angle between the axis and the slant) is given by $$\alpha = \arctan(r/h)$$. The full apex angle is $$2\alpha = 2\arctan(r/h)$$. For example, with $$r=5$$ and $$h=12$$, the half-apex angle is $$\arctan(5/12) \approx 22.6°$$.
The volume formula $$V = \frac{1}{3}\pi r^2 h$$ works for all cones (right or oblique) as long as $$h$$ is the perpendicular height from base to apex. However, the surface area and slant height formulas apply only to right circular cones where the apex is directly above the center of the base.
When unrolled, the lateral surface becomes a circular sector with radius $$l$$ (slant height) and arc length $$2\pi r$$. The sector angle is $$\theta = \frac{2\pi r}{l} \times \frac{180}{\pi} = \frac{360r}{l}$$ degrees. Cut this sector from flat material and roll it into a cone.
A frustum is a cone with the top cut off by a plane parallel to the base, producing two circular faces. Its volume is $$V = \frac{\pi h}{3}(R^2 + Rr + r^2)$$ where $$R$$ and $$r$$ are the two base radii and $$h$$ is the height. Frustums appear in bucket shapes, lampshades, and structural columns.
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The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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