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Cone Calculator

Computes the volume, slant height, base area, lateral (curved) surface area, total surface area and half-apex angle in degrees of a right circular cone from its base radius and height.

When to use

You need the capacity or surface of a funnel, cone-shaped pile, roof or any right circular cone.

Do not use when: The cone is truncated (frustum), oblique, or has a square base (use pyramid).

Formula

volume = π · radius² · height / 3; slant_height = √(radius² + height²); base_area = π · radius²; lateral_area = π · radius · slant_height; total_surface_area = π · radius · (radius + slant_height); half_apex_angle_deg = atan(radius / height)

Inputs

Parameter	Type	Unit	Required	Description
`radius`	number	units	yes	Radius of the circular base. Range: > 0
`height`	number	units	yes	Perpendicular distance from the base to the apex. Range: > 0

Outputs

Output	Type	Unit	Description
`volume`	number	units³	1/3 × π × r² × h.
`slant_height`	number	units	√(r² + h²), distance from the apex to the edge of the base.
`base_area`	number	units²	π × r².
`lateral_area`	number	units²	π × r × slant_height, the curved surface only.
`total_surface_area`	number	units²	π × r × (r + slant_height).
`half_apex_angle_deg`	number	degrees	Angle between the axis and the slant side = atan(r / h).

Example

Radius 3, height 4: {"radius":3,"height":4} → {"volume":37.6991,"slant_height":5,"base_area":28.2743,"lateral_area":47.1239,"total_surface_area":75.3982,"half_apex_angle_deg":36.8699}

Radius 5, height 12: {"radius":5,"height":12} → {"volume":314.1593,"slant_height":13,"lateral_area":204.2035,"total_surface_area":282.7433,"half_apex_angle_deg":22.6199}

GET https://tttkmbb.com/api/v1/calculate/cone?radius=3&height=4

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/cone (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/cone · Markdown: https://tttkmbb.com/geometry/cone.md · JSON definition: https://tttkmbb.com/geometry/cone.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="cone"
OpenAPI operationId: calculate_cone_properties
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Cone (reference)
Wolfram MathWorld – Cone (reference)

FAQ

Height or slant height?

Enter the perpendicular height (base to apex along the axis). If you only know the slant height l, the height is √(l² − r²).

Why is the volume a third of a cylinder's?

A cone with the same base and height has exactly one third of the cylinder's volume; this follows from integrating the shrinking cross-sections.

Related calculators

Cylinder Calculator — Same base without the taper (three times the volume).
Square Pyramid Calculator — The square-based counterpart.
Right Triangle Calculator — The cross-section through the axis is a right triangle with legs r and h.