Calculator
Cut a circular sector of this radius and angle from sheet metal, then roll and join the straight edges to form the cone.
Cone anatomy
Worked examples
Example 1 — full cone (r = 3, h = 4)
- Slant height: l = √(3² + 4²) = √25 = 5.
- Volume: V = π × 3² × 4 / 3 = 12π ≈ 37.6991.
- Lateral area: π × 3 × 5 = 15π ≈ 47.1239; base area: π × 9 ≈ 28.2743.
- Total surface area: π × 3 × (3 + 5) = 24π ≈ 75.3982.
Example 2 — frustum (R = 5, r = 3, h = 6)
- Slant height: s = √((5 − 3)² + 6²) = √40 ≈ 6.3246.
- Volume: V = π × 6 × (25 + 15 + 9) / 3 = 98π ≈ 307.8761.
- Lateral area: π × (5 + 3) × 6.3246 ≈ 158.9534.
- Total surface area (both circular ends included): ≈ 265.7676.
Example 3 — sheet-metal flat pattern (r = 10, h = 24)
- Slant height: l = √(10² + 24²) = √676 = 26.
- Cut a sector of radius 26 spanning 360° × 10 / 26 ≈ 138.4615°.
- Roll it up and join the straight edges — the arc (2π × 10 ≈ 62.8319) becomes the base circumference.
Frustum formulas (truncated cone)
Sanity check: with r = R the volume becomes π R² h (a cylinder); with r = 0 it becomes π R² h / 3 (a full cone).
Frequently asked questions
What is the formula for the volume of a cone?
The volume of a right circular cone is V = π r² h / 3, where r is the base radius and h is the vertical height. A cone holds exactly one-third the volume of a cylinder with the same base and height. For example, a cone with radius 3 cm and height 4 cm has volume π × 9 × 4 / 3 = 12π ≈ 37.6991 cm³.
How do you calculate the slant height of a cone?
The slant height l is the hypotenuse of the right triangle formed by the radius and the height: l = √(r² + h²). A cone with radius 3 and height 4 has slant height √(9 + 16) = 5 — the classic 3-4-5 triangle.
What is the formula for the surface area of a cone?
The lateral (side) surface area is π r l, and adding the circular base gives the total surface area π r (r + l), where l is the slant height. For radius 3 and height 4 (slant height 5), the total surface area is π × 3 × 8 = 24π ≈ 75.3982 square units.
What is the volume formula for a frustum of a cone?
A frustum (a cone with the top cut off) with bottom radius R, top radius r, and height h has volume V = π h (R² + R r + r²) / 3. Its slant height is √((R − r)² + h²) and its lateral area is π (R + r) × slant height.
How do you make a cone from sheet metal? (flat pattern)
Unfolded, a cone is a circular sector: the sector radius equals the cone's slant height l, and the sector angle is 360° × r / l. For example, a cone with radius 10 cm and height 24 cm has slant height 26 cm, so you cut a sector of radius 26 cm spanning 360 × 10 / 26 ≈ 138.46°.
Should I use radius or diameter in the cone formulas?
All standard cone formulas use the radius. If you measured the diameter, halve it first: r = d / 2. Using the diameter directly in V = π r² h / 3 overstates the volume by a factor of 4.
What units does the cone calculator use?
Any consistent length unit works — millimeters, centimeters, meters, inches, or feet. The volume comes out in the cube of that unit (for example cm → cm³), and areas in the square of that unit. The calculator also converts volumes to liters, US gallons, cubic inches, and cubic feet automatically.
How do I find the height of a cone from its volume?
Rearrange the volume formula: h = 3V / (π r²). Similarly, the radius from volume and height is r = √(3V / (π h)). The reverse-solver mode of this calculator does these rearrangements for you, including solving height from slant height (h = √(l² − r²)) or from total surface area.
What is the difference between a cone and a pyramid?
A cone has a circular base; a pyramid has a polygon base (triangle, square, and so on). Both follow the same one-third rule: volume = (base area × height) / 3. A square pyramid with base side s has volume s² h / 3.
What are real-world uses of cone calculations?
Funnels, traffic cones, ice-cream cones, conical hoppers and silos, loudspeaker cones, and pile volumes (sand, gravel, grain — where the angle of repose sets the height). Sheet-metal workers use the flat-pattern math to cut and roll cones, reducers, and hoppers.
What is a frustum?
A frustum is the part of a cone left after cutting off the top with a plane parallel to the base — like a bucket, lampshade, or plant-pot shape. Set the top radius equal to the bottom radius and the frustum formulas become the cylinder formulas; set the top radius to zero and they become the full-cone formulas.
Do these formulas account for wall thickness?
No — the formulas assume a solid cone or an infinitely thin shell. For a thick-walled hollow cone (for example a spun-metal funnel), compute the outer cone volume minus the inner cone volume, or multiply the lateral area by the wall thickness as a first approximation.
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