Lateral Area of a Cone Calculator
Calculate the lateral surface area, total surface area, and volume of a cone
Calculate Cone Lateral Area
Radius of the circular base of the cone
Perpendicular distance from base to apex
Cone Calculation Results
Formula used: A_L = π × r × √(r² + h²)
Slant height: 0 cm
Calculation: √(0² + 0²) = 0 cm
Example Calculation
Traffic Cone Example
Base radius (r): 6 cm
Vertical height (h): 10 cm
Formula: A_L = π × r × √(r² + h²)
Step-by-Step Solution
1. Calculate slant height: l = √(6² + 10²) = √(36 + 100) = √136 = 11.66 cm
2. Apply formula: A_L = π × 6 × 11.66
3. Calculate result: A_L = 3.1416 × 6 × 11.66 = 219.8 cm²
Final answer: 219.8 cm²
Cone Properties
Base Radius
Distance from center to edge of base
Height
Perpendicular distance from base to apex
Slant Height
Distance along surface from base edge to apex
Key Formulas
Lateral Area
A_L = π × r × l
A_L = π × r × √(r² + h²)
Total Surface Area
A_T = π × r × (r + l)
Volume
V = (1/3) × π × r² × h
Slant Height
l = √(r² + h²)
Understanding Lateral Area of a Cone
What is Lateral Area?
The lateral area of a cone is the curved surface area of the cone, excluding the circular base. It's the area you would get if you "unrolled" the curved surface of the cone into a flat sector of a circle.
Difference from Total Surface Area
- •Lateral Area: Only the curved surface (A_L = π × r × l)
- •Total Surface Area: Curved surface + base area (A_T = π × r × l + π × r²)
- •Base Area: Circular base only (A_B = π × r²)
Formula Derivation
Two Main Formulas:
A_L = π × r × l
A_L = π × r × √(r² + h²)
Where:
- • r = radius of the base
- • h = vertical height of cone
- • l = slant height = √(r² + h²)
- • π ≈ 3.14159
Note: The lateral area is exactly half the lateral area of a cylinder with the same radius and slant height as height.
Real-World Applications
Traffic Cones
Calculating material needed for manufacturing traffic cones
Ice Cream Cones
Determining wafer material for cone-shaped containers
Architectural Design
Calculating surface area for conical roofs and structures