Slant Height of a Cone Calculator
Calculate the slant height of a cone using radius, height, or angles with step-by-step solutions
Cone Slant Height Calculator
Results
Cone Properties
Cone Diagram
Common Examples
Ice Cream Cone
Traffic Cone
Quick Reference
Main Formula
With Base Angle
With Apex Angle
Key Concepts
Slant height is the hypotenuse of the cone's right triangle
Base angle α + apex angle β = 90°
Lateral area = π × r × l
Slant height is always longer than height
Applications
Architecture and construction
Manufacturing cone-shaped objects
Engineering and design
Educational geometry problems
Understanding Cone Slant Height
What is Slant Height?
The slant height of a cone is the distance from the apex (top point) to any point on the circumference of the base, measured along the surface of the cone. It represents the hypotenuse of the right triangle formed by the height and radius of the cone.
Main Formula
l = √(r² + h²)
This formula comes directly from the Pythagorean theorem, where the slant height is the hypotenuse, and the radius and height are the two legs of the right triangle.
Alternative Methods
When you know angles instead of both radius and height, you can use trigonometric relationships to find the slant height.
With Base Angle (α):
l = h / sin(α)
r = h / tan(α)
With Apex Angle (β):
l = r / sin(β)
h = r / tan(β)
Remember that the base angle α and apex angle β are complementary, meaning α + β = 90°.
Practical Applications
Construction
Calculating materials needed for cone-shaped roofs, silos, and towers
Manufacturing
Designing cone-shaped products like funnels, lamp shades, and containers
Education
Teaching geometry, trigonometry, and spatial visualization concepts