Cone Volume Calculator
Calculate the volume, surface area, lateral area, and slant height of a cone. Supports multiple solve modes, unit conversion, and real-world presets.
Slant Height Calculator — Cone, Pyramid & Frustum
Calculate the slant height, lateral surface area, total surface area, and volume of cones, pyramids (square, triangular, hexagonal), and frustums from base dimensions and vertical height.
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Planning notes, formulas, and examples
About the Slant Height Calculator — Cone, Pyramid & Frustum
The slant height of a 3D solid is the distance measured along the surface from the edge of the base to the apex (or, for a frustum, to the top edge). It is fundamentally different from the vertical height, which is the perpendicular distance from base to apex. The relationship between the two is governed by the Pythagorean theorem: slant height² = vertical height² + horizontal distance².
For a right circular cone with base radius r and vertical height h, the slant height is simply l = √(r² + h²). The lateral (side) surface area — the area of the cone's surface excluding the base — is πrl. For square pyramids, the horizontal distance is half the base side, so l = √((b/2)² + h²). Triangular and hexagonal pyramids use their respective apothems (the distance from the center to the midpoint of a base edge) instead.
Frustums — solids formed by cutting the top off a cone or pyramid with a horizontal plane — also have a slant height. For a conical frustum with radii R (bottom) and r (top), the slant height is l = √((R − r)² + h²). Frustums appear everywhere: buckets, lamp shades, cooling towers, and truncated pyramids.
This calculator supports six shape types: cone, square pyramid, triangular pyramid, hexagonal pyramid, conical frustum, and square frustum. For each, it computes the slant height, slope angle, lateral surface area, total surface area, base area, volume, and edge slant height. A Pythagorean-relationship visualization and a comprehensive formula reference table are included. Presets cover real-world objects from ice-cream cones to the Great Pyramid of Giza.
When This Page Helps
This calculator is useful because slant height is rarely the only quantity you need. In practice, once you know the geometry of a cone, pyramid, or frustum, you usually also need the lateral area for covering material, the total area for finishing, the volume for capacity, and the slope angle for fabrication or design. Having those values tied to the same input set reduces setup mistakes and makes it easier to compare different solids or design revisions.
How to Use the Inputs
- Select the solid type: cone, square pyramid, triangular pyramid, hexagonal pyramid, conical frustum, or square frustum.
- Choose a measurement unit (mm, cm, in, m, ft).
- Enter the base radius (for cones/conical frustums) or base side length (for pyramids/square frustums).
- For frustums, also enter the top radius or top side length.
- Enter the vertical height of the solid.
- Or click a preset to load a real-world example.
- View slant height, slope angle, lateral SA, total SA, volume, and more.
- Examine the Pythagorean relationship bar chart and the formula reference table.
Formula used
Cone: l = √(r² + h²), Lateral SA = πrl
Square Pyramid: l = √((b/2)² + h²), Lateral SA = 2bl
Triangular Pyramid: apothem a = b/(2√3), l = √(a² + h²)
Hexagonal Pyramid: apothem a = b√3/2, l = √(a² + h²)
Conical Frustum: l = √((R−r)² + h²), Lateral SA = π(R+r)l
Square Frustum: l = √((B−b)²/4 + h²), Lateral SA = 2(B+b)lExample Calculation
Result: Slant height ≈ 30.41 cm, lateral SA ≈ 3345.18 cm², volume ≈ 29059.73 cm³
For a conical frustum with bottom radius 20 cm, top radius 15 cm, and vertical height 30 cm, the slant height is √((20 − 15)² + 30²) = √925 ≈ 30.41 cm. The lateral surface area is π(20 + 15)(30.41) ≈ 3345.18 cm², and the frustum volume is (π × 30 / 3)(20² + 20 × 15 + 15²) ≈ 29059.73 cm³.
Tips & Best Practices
- The slant height is always longer than the vertical height — the Pythagorean theorem guarantees l > h when the horizontal distance is positive.
- For the Great Pyramid of Giza (base 230 m, height 146 m), the slant height is about 186 m — the distance you'd walk up one face.
- Frustum slant height is what matters for pouring molds — it determines how far molten material must travel along the surface.
- A 3-4-5 cone (r=3, h=4, l=5) is the simplest Pythagorean-triple cone and great for teaching.
- The slope angle equals arctan(r/h) for a cone. Steeper cones have larger angles and more lateral surface area.
One Idea, Several Solids
The unifying idea behind this calculator is that slant height comes from a right-triangle cross section. For a cone, the horizontal leg is the radius. For a square pyramid, it is half the base side. For triangular and hexagonal pyramids, it is the base apothem. For frustums, the horizontal leg becomes the difference between the large and small radii or half the difference between the large and small square sides. The formulas look different, but they are all variations of the same geometric structure.
Why Designers Care About Slant Height
Slant height is the measurement that controls the size of the side surface, so it matters whenever material is cut, wrapped, or fabricated. A metal cone, a paper cup, a lamp shade, and a truncated hopper all depend on the side distance rather than only the vertical rise. In those settings, lateral surface area is often more important than total surface area because it determines how much sheet material must be formed before bases or lids are attached.
Avoiding Common Interpretation Errors
A frequent mistake is to confuse vertical height, edge length, and slant height. They are only equal in very special cases. On pyramids, the slant height runs to the midpoint of a base edge, while the edge slant height runs to a base vertex and is therefore longer. On frustums, the top size matters just as much as the bottom size because the side surface leans according to the difference between the two openings. Comparing these quantities side by side helps build the correct spatial picture before moving into area or volume work.
Sources & Methodology
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Frequently Asked Questions
Slant height is the distance measured along the surface of a 3D solid from the base edge to the apex (or top edge for a frustum). It is different from vertical height, which is measured perpendicular to the base.
Use the formula l = √(r² + h²), where r is the base radius and h is the vertical height. This follows directly from the Pythagorean theorem.
For a square pyramid, slant height = √((b/2)² + h²), where b is the base side and h is the vertical height. The 'b/2' is the apothem of the square base.
A frustum is the portion of a solid (usually a cone or pyramid) that lies between two parallel planes cutting it. Think of it as a cone with the top cut off. Buckets, lamp shades, and cooling towers are frustum shapes.
Not for pyramids. Slant height goes from the base edge's midpoint to the apex (along a face). Edge length goes from a base vertex to the apex. Edge length ≥ slant height.
Lateral surface area is directly proportional to slant height. For a cone, LSA = πrl. For a square pyramid, LSA = 2bl. Longer slant heights mean more surface to cover.
More in this topic
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Pyramid Volume & Surface Area Calculator
Calculate the volume, lateral area, total surface area, and slant height of square, rectangular, and triangular pyramids. Includes famous pyramid presets and a reference table.