Slant Height Calculator
Our free circle calculator solves slant height problems. Get worked examples, visual aids, and downloadable results. Enter your values for instant results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
l = sqrt(h^2 + r^2)
For a cone, slant height l equals the square root of the sum of the squared height h and squared radius r. For a square pyramid, replace r with half the base edge length (a/2). The slant height forms the hypotenuse of a right triangle with the height and base apothem.
Worked Examples
Example 1: Cone Slant Height
Problem:Find the slant height of a cone with height 12 cm and base radius 5 cm.
Solution:Slant height l = sqrt(h^2 + r^2)\nl = sqrt(12^2 + 5^2)\nl = sqrt(144 + 25)\nl = sqrt(169) = 13 cm\nLateral surface area = pi * r * l = pi * 5 * 13 = 204.20 cm^2\nHalf-angle = arctan(5/12) = 22.62 degrees
Result:Slant Height = 13 cm | Lateral Area = 204.20 cm^2 | Half-angle = 22.62 deg
Example 2: Square Pyramid Slant Height
Problem:A square pyramid has height 10 m and base edge 8 m. Find its slant height.
Solution:Half base = a/2 = 8/2 = 4 m\nSlant height l = sqrt(h^2 + (a/2)^2)\nl = sqrt(100 + 16) = sqrt(116) = 10.77 m\nLateral area = 2 * a * l = 2 * 8 * 10.77 = 172.33 m^2\nBase area = 64 m^2\nTotal surface area = 172.33 + 64 = 236.33 m^2
Result:Slant Height = 10.77 m | Lateral Area = 172.33 m^2 | Total SA = 236.33 m^2
Frequently Asked Questions
What is slant height and how is it different from regular height?
Slant height is the distance measured along the lateral (side) surface of a three-dimensional shape from the apex to the base edge, while regular height (also called perpendicular height or altitude) is the vertical distance from the apex straight down to the center of the base. For a cone, the slant height runs along the surface from the tip to the circular base edge, while the height is the internal vertical distance from tip to base center. The slant height is always longer than the perpendicular height because it forms the hypotenuse of a right triangle where the height and radius are the two legs. Understanding this distinction is critical for calculating lateral surface area, which depends on slant height.
How do you calculate the slant height of a cone?
The slant height of a cone is calculated using the Pythagorean theorem because the slant height, perpendicular height, and radius form a right triangle. The formula is l = sqrt(h squared + r squared), where l is slant height, h is the perpendicular height, and r is the base radius. For example, a cone with height 12 and radius 5 has slant height = sqrt(144 + 25) = sqrt(169) = 13. This relationship works because if you slice a cone vertically through its apex, the cross-section is an isosceles triangle where the slant height is the equal sides, the height is the altitude, and the diameter is the base.
How do you find the slant height of a square pyramid?
For a square pyramid, the slant height is the distance from the apex to the midpoint of any base edge (not to a corner). It is calculated as l = sqrt(h squared + (a/2) squared), where h is the perpendicular height and a is the base edge length. The term a/2 represents the apothem of the square base, which is the distance from the center to the midpoint of an edge. For a pyramid with height 10 and base edge 8, the slant height = sqrt(100 + 16) = sqrt(116) = 10.77. Note that the lateral edge (from apex to corner) is different and longer: lateral edge = sqrt(h squared + (a*sqrt(2)/2) squared).
Why is slant height important for surface area calculations?
Slant height is essential for calculating the lateral (side) surface area of cones and pyramids because the lateral faces are not vertical but angled. For a cone, the lateral surface area equals pi * r * l, where l is the slant height. For a regular pyramid, the lateral area equals (1/2) * perimeter * slant height. Using the perpendicular height instead of slant height would underestimate the true surface area because the slanted surface covers more area than a vertical surface of the same height. This is practically important for determining material requirements such as the amount of fabric for a tent, metal for a funnel, or paint for a roof.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy