Calculate slant height, radius, or height of right circular cones instantly. Free cone calculator for geometry, engineering, and architecture with step-by-step examples.
The slant height of a cone is the distance from the apex (top point) of the cone to any point along the edge of its circular base. This cone slant height measurement is fundamental for calculating surface area, lateral surface area, and cone dimensions in geometry, engineering, and architecture.
Our cone slant height calculator enables you to find the slant height of a right circular cone when you know the radius and perpendicular height, or compute the radius or height from other known measurements. Whether you're working on geometry homework, engineering projects, or architectural designs, this tool provides accurate cone dimension calculations.
For a right circular cone, the slant height formula uses the Pythagorean theorem to calculate precise cone dimensions:
Where:
This formula arises because a right circular cone forms a right-angled triangle between the radius, height, and slant height.
You can rearrange the cone slant height formula to solve for radius or height in different scenarios:
To find the radius :
To find the height :
Zero or Negative Values: Radius, height, and slant height must be positive real numbers. Zero or negative values are not valid in the context of a physical cone. For example, a cone with or would be degenerate and not represent a valid three-dimensional shape.
Invalid Slant Height Values: The slant height must satisfy the condition and . If or , the cone cannot exist because the sides would not meet at a single apex.
Impossible Dimensions: If the calculated slant height is less than the radius or height, it's an indication of invalid dimensions. For instance, if units and units, the slant height must be greater than both 5 and 12 units due to the Pythagorean relationship.
Extremely Large Values: When dealing with very large numbers, be cautious of potential floating-point precision errors which could affect the accuracy of calculations.
Example 1: If units and units, the radius is negative, which is physically impossible. Adjust the value to a positive number.
Example 2: If units, units, and units, the dimensions are valid because and .
Example 3: If units, units, and units, the slant height is less than both the radius and height, which is impossible for a real cone.
Learn how to calculate cone dimensions with these detailed step-by-step examples:
Given:
Calculate the slant height ()
Given:
Calculate the radius ()
Given:
Calculate the height ()
Slant height calculations are essential in numerous professional and educational contexts:
While the slant height is crucial, sometimes other measures are more appropriate:
The study of cones dates back to ancient Greece. Mathematicians like Euclid and Apollonius of Perga made significant contributions to the understanding of conic sections. The concept of slant height arises from the Pythagorean theorem, attributed to Pythagoras (c. 570 – c. 495 BCE).
During the Renaissance, advancements in mathematics and engineering led to practical applications of these geometric principles in architecture and artisanship. The development of calculus further enhanced the ability to calculate properties of conic shapes with precision.
Today, the principles remain foundational in geometry and continue to have widespread applications in science, technology, engineering, and mathematics (STEM) fields.
An illustration of a right circular cone:
Here are code snippets in various programming languages to calculate the slant height:
1=SQRT(A2^2 + B2^2)
2Assuming A2 contains the radius and B2 contains the height.
1import math
2
3def slant_height(r, h):
4 return math.hypot(r, h)
5
6## Example usage
7radius = 5
8height = 12
9print(f"Slant Height: {slant_height(radius, height)}")
101function slantHeight(r, h) {
2 return Math.hypot(r, h);
3}
4
5// Example usage
6const radius = 5;
7const height = 12;
8console.log("Slant Height:", slantHeight(radius, height));
91public class Cone {
2 public static double slantHeight(double r, double h) {
3 return Math.hypot(r, h);
4 }
5
6 public static void main(String[] args) {
7 double radius = 5;
8 double height = 12;
9 System.out.println("Slant Height: " + slantHeight(radius, height));
10 }
11}
121using System;
2
3class Cone
4{
5 static double SlantHeight(double r, double h)
6 {
7 return Math.Sqrt(r * r + h * h);
8 }
9
10 static void Main()
11 {
12 double radius = 5;
13 double height = 12;
14 Console.WriteLine("Slant Height: " + SlantHeight(radius, height));
15 }
16}
171function l = slantHeight(r, h)
2 l = hypot(r, h);
3end
4
5% Example usage
6radius = 5;
7height = 12;
8disp(['Slant Height: ', num2str(slantHeight(radius, height))]);
91slant_height <- function(r, h) {
2 sqrt(r^2 + h^2)
3}
4
5## Example usage
6radius <- 5
7height <- 12
8cat("Slant Height:", slant_height(radius, height), "\n")
91package main
2
3import (
4 "fmt"
5 "math"
6)
7
8func slantHeight(r, h float64) float64 {
9 return math.Hypot(r, h)
10}
11
12func main() {
13 radius := 5.0
14 height := 12.0
15 fmt.Printf("Slant Height: %.2f\n", slantHeight(radius, height))
16}
171def slant_height(r, h)
2 Math.hypot(r, h)
3end
4
5## Example usage
6radius = 5
7height = 12
8puts "Slant Height: #{slant_height(radius, height)}"
91<?php
2function slantHeight($r, $h) {
3 return sqrt($r * $r + $h * $h);
4}
5
6// Example usage
7$radius = 5;
8$height = 12;
9echo "Slant Height: " . slantHeight($radius, $height);
10?>
111fn slant_height(r: f64, h: f64) -> f64 {
2 (r.powi(2) + h.powi(2)).sqrt()
3}
4
5fn main() {
6 let radius = 5.0;
7 let height = 12.0;
8 println!("Slant Height: {}", slant_height(radius, height));
9}
101import Foundation
2
3func slantHeight(_ r: Double, _ h: Double) -> Double {
4 return sqrt(r * r + h * h)
5}
6
7// Example usage
8let radius = 5.0
9let height = 12.0
10print("Slant Height: \(slantHeight(radius, height))")
11The slant height of a cone is the distance from the apex (tip) to any point on the edge of the circular base, measured along the surface of the cone.
Use the formula l = √(r² + h²) where l is slant height, r is radius, and h is height. This applies the Pythagorean theorem to cone geometry.
The height is the perpendicular distance from base to apex, while slant height is measured along the cone's surface from apex to base edge.
No, the slant height must always be greater than both radius and height due to the Pythagorean relationship in cone geometry.
You can use any consistent units (inches, centimeters, meters, feet) as long as all measurements use the same unit system.
Slant height is essential for calculating lateral surface area, total surface area, and determining material requirements in manufacturing and construction.
Our calculator provides highly accurate results using precise mathematical formulas, suitable for professional engineering and educational applications.
This calculator is designed specifically for right circular cones. Oblique cones require different geometric approaches.
Use our slant height of cone calculator to solve geometry problems, complete engineering projects, or tackle architectural challenges. Simply enter your known measurements to get instant, accurate results for all your cone dimension calculations.
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