Calculate volume, surface area, and dimensions of a right circular cone. Enter radius with height, slant height, or volume to find all properties.
Last updated: April 2026 | By Patchworkr Team
A right circular cone is a 3D shape with a circular base and a point (apex) directly above the center of the base. It's one of the fundamental geometric solids.
Key measurements:
Cones appear everywhere: ice cream cones, traffic cones, party hats, funnels, and in nature as volcanic mountains and tornado funnels.
Determine which measurements you have available: radius (r), height (h), slant height (s), or volume (V). Why: Different combinations require different calculation approaches. You don't need all measurements—any valid pair will give you all the rest. This determines which formula path to use.
Why: The slant height connects to both radius and height through the Pythagorean theorem. If you have r and h, calculate s. If you have r and s, calculate h = √(s² - r²). The slant height is essential for surface area and lateral calculations. It represents the shortest path on the cone's surface from base to apex.
Why: This is the fundamental volume formula. A cone with the same base area and height as a cylinder holds exactly 1/3 of that cylinder's volume. This relationship emerges from calculus integration over circular cross-sections. For any real application (storage, packaging, ingredients), volume calculation is the primary goal.
Why: Surface area determines material needed for covering or construction. Base area is a simple circle calculation. Lateral area represents the cone's curved surface—when "unrolled," it forms a sector of a circle with radius s. The distinction matters for applications like paint, sheet metal, or thermal analysis where you might only cover the sides.
Once volume and surface areas are calculated, verify consistency: check that s² = r² + h², that all values are positive, and that volume makes geometric sense relative to dimensions. Document radius, height, slant height, volume, base area, lateral area, and total surface area. Why: Complete property documentation prevents mistakes and provides all information needed for engineering specs, material purchasing, or quality verification. Cross-checking relationships catches calculation errors early.
Ice Cream Cone
V = (πr²h)/3, where r is the base radius and h is the height. It's exactly one-third the volume of a cylinder with the same dimensions.
Use the Pythagorean theorem: s = √(r² + h²). The slant height, radius, and height form a right triangle.
Height (h) is vertical from base to apex. Slant height (s) is along the surface from base edge to apex. Always s > h.
Total surface area = πr² + πrs = πr(r + s). That's the circular base plus the lateral (curved) surface.
This can be proven with calculus or demonstrated physically: three identical cones of water fill one cylinder of the same base and height.
Yes, that's called an elliptic cone. But 'cone' usually means a circular cone unless specified otherwise.
A frustum is a cone with the top cut off parallel to the base, creating two circular faces. Like a bucket or lampshade.
Cones calculate volumes for silos, hoppers, funnels, conical tanks, traffic cones, and even volcanic crater volumes.