Interactive 3D Cone Diagram & Geometry Visualizer
Rotate, zoom, and inspect a right circular cone in real-time 3D space. Switch between shaded perspective and 2D cross-section cutaways to examine the relationship between radius ($r$), vertical altitude ($h$), slant height ($l$), and apex half-angle ($\theta$).
Dimension Controls
Adjust the cone dimensions to observe real-time geometric scaling.📐 Geometric Properties Inspector
- Slant Height ($l = \sqrt{r^2 + h^2}$): 13.0000 cm
- Base Area ($B = \pi r^2$): 78.5398 cm²
- Lateral Area ($A_{lat} = \pi r l$): 204.2035 cm²
- Total Area ($A = \pi r(r + l)$): 282.7433 cm²
- Volume ($V = \frac{1}{3}\pi r^2 h$): 314.1593
Anatomy of a Right Circular Cone
Understanding the foundational definitions and geometric elements of conical solids.
1. Apex (Vertex)
The single point at the sharp top of the cone that lies opposite the circular base. All straight line segments on the cone's lateral surface converge at the apex.
2. Base & Radius ($r$)
The flat planar bottom surface, which is a perfect circle in a right circular cone. The radius $r$ is the distance from the center of the base circle to its perimeter ($C = 2\pi r$).
3. Vertical Altitude ($h$)
The perpendicular distance from the apex straight down to the base plane. In a right cone, the line segment connecting the apex to the center of the base forms a strict $90^\circ$ right angle.
4. Slant Height ($l$ / Generatrix)
The distance measured along the outside surface from the apex to any point on the base perimeter. By the Pythagorean theorem: $l = \sqrt{r^2 + h^2}$.
5. Right Cone vs. Oblique Cone
In a right cone, the apex aligns directly above the center of the base. In an oblique cone, the apex is tilted or displaced off-center. By Cavalieri's principle, both possess identical volume if their base area and vertical height match!
6. Apex Semi-Vertical Angle ($\theta$)
The angle between the central vertical axis and the slant edge: $\tan(\theta) = \frac{r}{h}$. The total opening cone angle is $2\theta$.