Slant Height Calculator
Enter the vertical height and the base dimension of a right circular cone or a right square pyramid to find its slant height. Switch to reverse-solve mode to work backwards from the slant height to find the vertical height. Results update as you type, with a step-by-step breakdown of the Pythagorean theorem calculation.
Formula
Worked example
A cone has vertical height h = 12 cm and base radius r = 5 cm. The slant height l = sqrt(12² + 5²) = sqrt(144 + 25) = sqrt(169) = 13 cm. For a square pyramid with h = 12 cm and base side a = 10 cm: apothem = 10/2 = 5 cm, l = sqrt(12² + 5²) = 13 cm.
What is slant height?
Slant height is the straight-line distance measured along the lateral surface of a cone or pyramid from the apex (tip) down to the midpoint of a base edge. It is different from the vertical height, which is the perpendicular distance from the apex straight down to the centre of the base. For a right circular cone, the slant height runs from the apex to any point on the rim of the circular base. For a right square pyramid, it runs from the apex to the midpoint of any of the four base edges. Both cases create a right triangle with the vertical height and the apothem (half the base side for a pyramid, or the radius for a cone) as the two legs, making slant height the hypotenuse of that triangle.
The formula and how it works
Because the vertical height, the apothem (or radius), and the slant height form a right-angled triangle, the Pythagorean theorem applies directly. For a cone: l = sqrt(h² + r²), where h is the vertical height and r is the base radius. For a square pyramid: l = sqrt(h² + (a/2)²), where a is the side length of the square base and a/2 is the apothem. The same equation can be rearranged to solve for any unknown: if you know the slant height and the base dimension, subtract r² from l² and square-root to find h; if you know the slant height and height, subtract h² from l² and square-root to find r (then double r for the full base side of a pyramid). This calculator handles all three modes automatically.
Why slant height matters: surface area
Slant height is the key measurement for computing lateral surface area. For a cone, the curved surface area equals pi times the radius times the slant height (A = pi*r*l). Adding the circular base gives the total surface area: pi*r*(l + r). For a square pyramid, the lateral surface area is 2*a*l, which accounts for four triangular faces each with base a and height l. The total surface area adds the square base: 2*a*l + a². Without the slant height you cannot calculate how much material you need to cover or paint the surface, which makes it an essential quantity in packaging, architecture, manufacturing, and geometry coursework.
Slant height vs. lateral edge
Slant height is sometimes confused with the lateral edge. The lateral edge of a pyramid is the line from the apex to a corner of the base (a vertex), whereas the slant height goes from the apex to the midpoint of a base edge. The lateral edge is always longer than the slant height for a square pyramid. For a right square pyramid with base side a: lateral edge d = sqrt(h² + (a*sqrt(2)/2)²) = sqrt(h² + a²/2). The slant height is l = sqrt(h² + (a/2)²) = sqrt(h² + a²/4). Because a²/2 > a²/4, the lateral edge is always larger than the slant height, and only the slant height is used in the surface area formula.
Common slant height examples
| Shape | Height (h) | Base dimension | Slant height (l) | Notes |
|---|---|---|---|---|
| Cone | 12 | r = 5 | 13 | Classic 5-12-13 Pythagorean triple |
| Cone | 8 | r = 6 | 10 | Classic 6-8-10 triple (2× of 3-4-5) |
| Cone | 4 | r = 3 | 5 | 3-4-5 Pythagorean triple |
| Square pyramid | 12 | a = 10, apothem = 5 | 13 | Apothem = 5, uses 5-12-13 triple |
| Square pyramid | 8 | a = 12, apothem = 6 | 10 | Apothem = 6, uses 6-8-10 triple |
| Great Pyramid (Giza approx.) | 138.8 m | a = 230.4 m, apothem = 115.2 m | 179.7 m | Historical approximation |
Illustrative cones and pyramids with h and base given. All dimensions in the same unit.
Frequently asked questions
What is the difference between slant height and vertical height?
Vertical height (h) is the perpendicular distance from the apex straight down to the centre of the base, measured inside the solid. Slant height (l) is measured along the outside surface from the apex to the midpoint of a base edge. For any non-degenerate cone or pyramid, the slant height is always greater than the vertical height because it is the hypotenuse of a right triangle whose other two sides are h and the apothem.
Can I use the same formula for a cone and a pyramid?
Yes, with one small difference. For a cone you use the base radius r directly: l = sqrt(h² + r²). For a square pyramid you use the apothem, which is half the base side a: l = sqrt(h² + (a/2)²). In both cases the slant height is the hypotenuse of a right triangle whose legs are the vertical height and the apothem/radius.
How do I find the slant height if I only know the surface area?
Rearrange the surface area formula. For a cone with known radius r and lateral area A: l = A / (pi * r). For a pyramid with known base side a and lateral area A: l = A / (2 * a). Once you have l and one of h or r, you can find the remaining dimension with l = sqrt(h² + r²).
What is the apothem of a pyramid?
The apothem is the distance from the centre of the base to the midpoint of one of its edges. For a square pyramid with base side a, the apothem is a/2. It is the horizontal leg of the right triangle formed by the vertical height, the apothem, and the slant height. Do not confuse the apothem with the lateral edge, which runs from the apex to a corner (vertex) of the base.
What is the slant height of the Great Pyramid of Giza?
The original Great Pyramid of Giza had a base side of approximately 230.4 m and a vertical height of approximately 146.5 m (it is now slightly shorter due to erosion). The apothem is 230.4/2 = 115.2 m, giving a slant height of sqrt(146.5² + 115.2²) = sqrt(21462.25 + 13271.04) = sqrt(34733.29) which is approximately 186.4 m for the original height. Modern survey estimates vary slightly.