Slant Height
Calculate the slant height of a right circular cone or right square pyramid using the Pythagorean theorem l = √(r² + h²). Free online geometry tool for 3D shape measurement problems.
About This Calculator
Slant height is a fundamental measurement in three-dimensional geometry, representing the distance along the sloping outer surface from the base edge to the apex of a cone or pyramid. It differs from the vertical height (altitude), which is the perpendicular distance through the interior of the shape. Understanding slant height is essential for calculating surface areas and material requirements for conical and pyramidal objects in construction, manufacturing, and education.
Our slant height calculator supports two common 3D shapes. For a right circular cone, use the formula l = \u221a(r\u00b2 + h\u00b2) where r is the base radius and h is the vertical height. For a right square pyramid, use l = \u221a((b/2)\u00b2 + h\u00b2) where b is the full base side length. In both cases, the radius (or half the base width) and height form a right triangle with the slant height as the hypotenuse. Simply select your shape type and enter your measurements in consistent units to get the slant height instantly.
This calculator is invaluable for students learning solid geometry, craftspeople building conical or pyramidal structures, teachers creating lesson demonstrations, and professionals needing quick slant height values for surface area calculations. The shareable URL feature preserves your shape selection and input values, letting you save and share specific calculations with ease.
Frequently Asked Questions
What is the formula for slant height?
The slant height (l) of a right circular cone or regular pyramid is calculated using the Pythagorean theorem: l = \u221a(r\u00b2 + h\u00b2), where r is the base radius (for a cone) or half the base width (for a pyramid), and h is the vertical height. For a cone the formula uses l = \u221a(r\u00b2 + h\u00b2) with the base radius; for a square pyramid use l = \u221a((b/2)\u00b2 + h\u00b2) where b is the full base side length.
What is the difference between slant height and vertical height?
Vertical height (or altitude) is the perpendicular distance from the base to the apex measured straight up through the center of the shape. Slant height is the distance along the sloping outer surface from the edge of the base to the apex. For both cones and pyramids, the slant height is always longer than the vertical height.
How do you calculate the slant height of a cone?
To calculate the slant height of a right circular cone, square the base radius (r), square the vertical height (h), add them together, and take the square root: l = \u221a(r\u00b2 + h\u00b2). For example, a cone with radius 5 cm and height 10 cm has a slant height of \u221a(25 + 100) = \u221a125 \u2248 11.18 cm.
How do you calculate the slant height of a square pyramid?
For a right square pyramid with base side length b and vertical height h, first divide the base width by 2, then apply the Pythagorean theorem: l = \u221a((b/2)\u00b2 + h\u00b2). For example, a pyramid with base width 8 cm and height 10 cm has a slant height of \u221a(16 + 100) = \u221a116 \u2248 10.77 cm.
Why is slant height important in geometry?
Slant height is critical for calculating the lateral surface area of cones (A = \u03c0rl) and pyramids (A = \u00bd \u00d7 perimeter \u00d7 slant height), for determining material needed to construct conical or pyramidal objects, and for engineering analysis of tapered structures. It is a prerequisite input for most cone and pyramid surface area formulas.
Is slant height the same for a cone and a pyramid?
The concept is the same -- both use the Pythagorean theorem where the base dimension and vertical height form the legs of a right triangle. However, a cone uses the base radius while a pyramid uses half the base width. The calculator supports both shapes with the appropriate formulas.
What units does the slant height calculator use?
The calculator works with any consistent linear unit. Enter base dimension and height in the same unit (cm, m, in, ft), and the slant height result will be in the same unit. For example, centimeter inputs produce a slant height in centimeters.
Can I calculate slant height for irregular pyramids or cones?
This formula applies to right circular cones (apex directly above the center of a circular base) and right square pyramids (apex above the center of a square base). For irregular or oblique shapes, more complex calculations are needed that account for the off-center apex position.