How to Calculate Slant Height
Slant height (l) is the distance measured along the outside of a cone or pyramid, from the apex down to the edge of the base — as opposed to the height (h), which is the straight vertical (perpendicular) distance from the apex to the center of the base. Because the apex, the base center, and a point on the base edge always form a right triangle, slant height is found with the Pythagorean theorem: l² = h² + (base measurement)². This calculator solves that relationship for a right circular cone and for a right pyramid with a square base, and also reports lateral surface area, total surface area, and volume.
Cone slant height formula
For a right circular cone with base radius r and height h, the slant height is l = √(r² + h²). The right triangle has legs r (base radius) and h (height), with the slant height as the hypotenuse running along the cone's curved surface. Once you have l, the lateral (side) surface area is π r l, and the total surface area adds the circular base: π r² + π r l = π r (r + l). Volume is (1/3) π r² h — note that slant height does not appear in the volume formula, since volume depends only on the base area and the perpendicular height.
Square pyramid slant height formula
For a right pyramid with a square base of side length s and height h, the slant height along each triangular face is l = √(h² + (s/2)²). Here the right triangle runs from the apex, straight down to the base center (length h), then out to the midpoint of a base edge (length s/2, the apothem of the square base) — the hypotenuse of that triangle is l. Lateral surface area is the sum of the four triangular faces: 2 s l (equivalently, ½ × perimeter × l = ½ × 4s × l). Total surface area adds the square base: s² + 2 s l. Volume is (1/3) s² h.
Common sources of error
- Confusing height with slant height: h is the vertical distance to the base center; l is the longer, slanted distance to the base edge. Do not use one where the formula calls for the other.
- Wrong base measurement for a pyramid: the pyramid slant-height formula uses half the base side (s/2), not the full side length s — using s instead of s/2 will overstate the result.
- Unit mismatch: enter the base measurement and height in the same unit; convert first if your measurements come from different unit systems.
Checking your result and applications
A quick sanity check: slant height must always be greater than both the height and the base measurement it is paired with (r, or s/2) — since it is the hypotenuse of a right triangle, it is the longest side. If your slant height comes out smaller than h or r/(s/2), recheck your inputs. Slant height calculations show up in roofing (a hip or pyramid roof's rafter length), tent and canopy design, funnel and hopper fabrication, traffic cone and lampshade manufacturing, and geometry problems involving surface area or material estimates for conical or pyramidal shapes.