Why does a cone have exactly one-third the volume of a cylinder?

In solid geometry, a cone with a base radius (r) and height (h) fits exactly three times inside a cylinder of identical radius and height. This ratio of 1:3 holds true for any pyramid compared to its prism counterpart.

What is slant height?

Slant height (l) is the shortest straight-line distance along the curved surface of the cone from the outer circular rim to the top apex. It forms a right triangle with the radius and height.

How do you find the surface area of a cone?

The total surface area is the sum of the flat circular base (Area = πr²) and the curved side wrap (Lateral Area = πrl, where l is slant height): Total Area = πr² + πrl.

Geometry Tool

Cone Calculator

Calculate the volume, surface area, base area, lateral area, and slant height of a right circular cone given its base radius and height.

Interactive Cone Calculator Workspace

Live Solver

Base Radius (r)

Vertical Height (h)

Solved Cone Properties

Volume (V)314.1593

Slant Height (l)13.0000

Total Area282.7433

Lateral Area204.2035

Step-by-Step Calculation

1. Slant Height (l): √(r² + h²) = √(5² + 12²) = √(25 + 144) = 13.0000

2. Base Circular Area: πr² = 3.14159 × 5² = 78.5398

3. Lateral Area: π × r × l = 3.14159 × 5 × 13.0000 = 204.2035

4. Volume: 1/3 × Base Area × h = 1/3 × 78.5398 × 12 = 314.1593

Mathematical Formula & Variables

Equation ModelVolume = 1/3 × π × r² × h, Slant Height (l) = √(r² + h²)

Solves the internal volume, flat base circular area, slanted height, and outer curved lateral surface area of a right circular cone.

Variable Definitions

Symbol	Description
r (Radius)	The radius of the circular base at the bottom of the cone.
h (Height)	The vertical distance from the center of the base to the apex point.
l (Slant Height)	The linear distance from any boundary edge point on the circular base to the apex.
V (Volume)	The internal 3D capacity (exactly one-third of a cylinder with identical dimensions).

How to Use the Cone Calculator

Enter the Base Radius of your cone.
Enter the vertical Height of your cone.
Check the instantly computed Slant Height, Volume, Base Area, Lateral Area, and Total Area.
View the steps to see how the Pythagorean Theorem solves the slant height (l = √(r² + h²)).

Practical Example Calculation

Scenario Context: A concrete funnel shaped like an inverted cone is built with a base radius of 5 meters and a height of 12 meters.

Step 1: Identify given values: Radius (r) = 5 m, Height (h) = 12 m.

Step 2: Calculate Slant Height (l): l = √(r² + h²) = √(5² + 12²) = √(25 + 144) = √169 = 13 meters.

Step 3: Calculate Volume (V): V = 1/3 × π × r² × h = 1/3 × π × 25 × 12 = 100 × π ≈ 314.16 cubic meters.

Step 4: Calculate Base Area (BA): BA = π × r² = π × 25 ≈ 78.54 square meters.

Step 5: Calculate Lateral Area (LA): LA = π × r × l = π × 5 × 13 = 65 × π ≈ 204.20 square meters.

Step 6: Calculate Total Area (TA): TA = BA + LA = 78.54 + 204.20 = 282.74 square meters.

The funnel volume is 314.16 m³, slant height is 13.00 m, and total outer surface is 282.74 m².

Frequently Asked Questions

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