About

In geometry and engineering, the cone is a common shape found in funnels, hoppers, and architectural spires. Determining the vertical height (altitude) is essential for calculating volume or ensuring that a part fits within a specific vertical clearance. Since the height is often internal and difficult to measure directly, it is usually calculated from external dimensions.

This tool offers two modes of calculation: determining height from the Slant Height (using the Pythagorean Theorem) or from the Volume. It includes a responsive wireframe visualization that adjusts its proportions to match your inputs, giving you immediate visual confirmation of the cone's geometry.

Formulas

For a right circular cone, the Radius (r), Height (h), and Slant Height (l) form a right-angled triangle. By the Pythagorean theorem:

l² = r² + h²

Rearranging for Height:

h = √l² - r²

Reference Data

Parameter	Formula	Description
Height (from Slant)	h = √l² - r²	Right triangle logic.
Height (from Volume)	h = 3Vπr²	Derived from Volume formula.
Volume	V = 1/3πr²h	Space occupied.
Slant Height	l	Distance from apex to base edge.

Frequently Asked Questions

The slant height is the distance from the tip (apex) of the cone down the side to any point on the edge of the circular base. It is always longer than the vertical height.

In a right triangle, the hypotenuse (slant height) must be the longest side. If the radius were equal to or larger than the slant height, the triangle (and thus the cone) could not exist physically.

No, this calculator assumes a 'Right Circular Cone', where the apex is directly above the center of the base. Oblique cones require more complex trigonometry.

The wireframe model is a proportional representation generated using HTML5 Canvas. It scales based on your inputs to show the relative difference between width and height.