Right Circular Cone Calculator
Calculate volume, surface area, lateral area, and base area of a right circular cone
Enter Cone Dimensions
Distance from center to edge of base
Vertical distance from base to apex
Distance from base edge to apex
Cone Properties
All Dimensions
Example Calculation
Ice Cream Cone Example
Given:
Base radius (r) = 3 cm
Height (h) = 4 cm
Step-by-Step Solution
1. Slant Height: l = √(3² + 4²) = √(9 + 16) = √25 = 5 cm
2. Volume: V = (1/3) × π × 3² × 4 = (1/3) × π × 9 × 4 = 12π ≈ 37.7 cm³
3. Base Area: A_B = π × 3² = 9π ≈ 28.3 cm²
4. Lateral Area: A_L = π × 3 × 5 = 15π ≈ 47.1 cm²
5. Total Surface Area: A = 28.3 + 47.1 = 75.4 cm²
Key Formulas
Volume
V = (1/3)πr²h
One-third of base area × height
Total Surface Area
A = πr(r + l)
Base area + lateral area
Lateral Surface Area
A_L = πrl
Curved side surface only
Slant Height
l = √(r² + h²)
Using Pythagorean theorem
Cone Properties
Circular base with apex directly above center
Slant height forms hypotenuse of right triangle
Height is perpendicular distance to apex
Lateral surface can be unfolded into a sector
Volume is 1/3 that of corresponding cylinder
Understanding Right Circular Cones
What is a Right Circular Cone?
A right circular cone is a three-dimensional shape with a circular base and an apex (vertex) directly above the center of the base. The "right" designation means the apex is positioned perpendicular to the base, making the height a straight vertical line from the base center to the apex.
Key Components
- •Base: Circular flat surface at the bottom
- •Apex/Vertex: Point at the top of the cone
- •Height: Perpendicular distance from base to apex
- •Slant Height: Distance from base edge to apex
- •Radius: Distance from base center to edge
Real-World Applications
Engineering & Architecture
Traffic cones, roofing, funnels, and conical tanks for liquid storage
Food Industry
Ice cream cones, pastry bags, paper cups, and food packaging
Manufacturing
Material volume calculations, packaging design, and container optimization
Education
Teaching 3D geometry, volume relationships, and surface area concepts
Mathematical Relationships
Pythagorean Relationship
The radius (r), height (h), and slant height (l) form a right triangle:
l² = r² + h²
Volume Relationship
A cone's volume is exactly one-third that of a cylinder with the same base and height:
V_cone = (1/3) × V_cylinder