What the Cone Slant Height Calculator Does
The Cone Slant Height Calculator works out the distance from the tip (apex) of a right circular cone down to the edge of its circular base — a measurement known as the slant height. You only need two values: the cone's vertical Height and its base Radius. From these, the tool instantly returns the slant height, and as a useful bonus it also calculates the cone's total surface area and volume.
The Inputs You Enter
- Height: the perpendicular distance from the centre of the base straight up to the apex.
- Radius: the distance from the centre of the circular base to its edge.
Use the same unit for both fields (for example centimetres or inches), and the results will be returned in that same unit — with surface area squared and volume cubed.
The Formula Explained
The slant height is found using the Pythagorean theorem, because the height, radius and slant height form a right triangle:
l = √(h² + r²)
Here h is the height and r is the radius. The calculator also computes:
- Surface area = π × r × (r + √(h² + r²)) — base plus curved side.
- Volume = (1/3) × π × r² × h.
Worked Example
Suppose a cone has a height of 4 and a radius of 3.
- Slant height: l = √(4² + 3²) = √(16 + 9) = √25 = 5
- Surface area: π × 3 × (3 + 5) = π × 3 × 8 ≈ 75.40
- Volume: (1/3) × π × 3² × 4 = (1/3) × π × 36 ≈ 37.70
So a cone 4 units tall with a 3-unit radius has a slant height of exactly 5 units.
Frequently Asked Questions
What is the difference between height and slant height? The height is the straight vertical distance from base to apex. The slant height runs along the sloped outer surface from the base edge to the apex, so it is always longer than the height.
Can the slant height ever be shorter than the height or radius? No. Because it is the hypotenuse of a right triangle, the slant height is always greater than both the height and the radius individually.
Why does the calculator also show surface area and volume? The slant height is a building block for the surface area formula, so calculating all three together saves time on geometry homework, design and 3D modeling tasks.
