Cone Calculator
Calculate cone volume, lateral surface area, total surface area, and slant height. Includes frustum (truncated cone), cone angle, and weight from density.
cm
cm
Volume
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Slant Height —
Lateral Surface Area —
Total Surface Area —
Extended More scenarios, charts & detailed breakdown ▾
Volume (cm³)
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Lateral SA (cm²) —
Slant Height (cm) —
Professional Full parameters & maximum detail ▾
Angles & Geometry
Apex Half-Angle (°) —
Full Apex Angle (°) —
Centroid Height from Base (cm) —
Weight & Pattern
Weight (g) —
Unrolled Sector Angle (°) —
How to Use This Calculator
- Enter Radius and Height to get volume, surface areas, and slant height.
- Use the From Slant Height tab if you know slant height instead of height.
- Use the Frustum tab for truncated cone with two radii.
- The Professional tab adds apex angle, centroid, weight, and the unrolled flat pattern sector angle.
Formula
Volume: V = (1/3)πr²h | Slant: l = √(r²+h²) | LSA: πrl | TSA: πr(l+r)
Example
r=4 cm, h=9 cm → l ≈ 9.85 cm, V ≈ 150.80 cm³, TSA ≈ 173.78 cm².
Frequently Asked Questions
- The volume of a cone is V = (1/3)πr²h, where r is the base radius and h is the vertical height. For a cone with r = 4 cm and h = 9 cm: V = (1/3) × π × 16 × 9 ≈ 150.80 cm³. A key insight: a cone holds exactly one-third of the volume of a cylinder with the same base radius and height. This factor of 1/3 arises because a cone tapers to a point, whereas a cylinder maintains its full cross-section throughout its height.
- Slant height (l) is the distance measured along the surface of a cone from the apex (tip) to any point on the base circle edge. It is computed using the Pythagorean theorem: l = √(r² + h²), where r is the base radius and h is the vertical height. For r = 4 cm, h = 3 cm: l = √(16 + 9) = √25 = 5 cm. Slant height is used in the lateral surface area formula (LSA = πrl) and to unroll the cone into a flat sector for fabrication or packaging design.
- A frustum is the portion of a cone that remains after cutting off the top with a plane parallel to the base. It has a larger base radius R₁, a smaller top radius R₂, and a vertical height h. Frustum volume = (πh/3)(R₁² + R₁R₂ + R₂²). For example, R₁ = 5, R₂ = 3, h = 4: V = (π × 4/3)(25 + 15 + 9) = (4π/3)(49) ≈ 205.25 cubic units. Frustums appear in buckets, drinking glasses, lampshades, and many engineering components.
- The half-angle (apex semi-angle) θ of a cone is the angle between the axis (centerline) and the slant side. It is calculated as θ = arctan(r/h), where r is the base radius and h is the height. The full apex angle = 2θ. Example: r = 4 cm, h = 3 cm → θ = arctan(4/3) ≈ 53.13°, apex angle ≈ 106.26°. A sharper, taller cone has a smaller half-angle; a flatter, wider cone has a larger half-angle. A half-angle of 45° means r = h.
- The lateral surface area (LSA) of a cone is the area of the curved surface only, excluding the circular base. Formula: LSA = πrl, where r is the base radius and l = √(r² + h²) is the slant height. The total surface area (TSA) includes the base: TSA = πrl + πr² = πr(l + r). Example: r = 4 cm, h = 3 cm → l = 5 cm → LSA = π × 4 × 5 ≈ 62.83 cm², TSA = π × 4 × (5 + 4) ≈ 113.10 cm². The LSA equals the area of a circular sector when the cone is unrolled flat.