🍦 Right Circular Cone Calc: find A, V, A_L, A_B
Calculate all properties of a right circular cone
How to Use This Calculator
Enter Radius and Height
Input the radius of the circular base and the perpendicular height of the cone. Make sure both values are positive numbers.
Click Calculate
Press the "Calculate All Properties" button to compute the total surface area (A), volume (V), lateral area (A_L), base area (A_B), and slant height.
Review All Results
View all calculated properties: total surface area, volume, lateral area (curved surface), base area (circular base), and slant height. The calculator shows step-by-step formulas for each.
Formulas
Slant Height: s = √(r² + h²)
Distance from edge to apex along the surface
Volume: V = (1/3)πr²h
One-third the volume of a cylinder with same base and height
Lateral Area: A_L = πrs
Area of the curved surface (excluding base)
Base Area: A_B = πr²
Area of the circular base
Total Surface Area: A = A_L + A_B = πrs + πr² = πr(s + r)
Sum of lateral area and base area
Where:
- r = radius of the circular base
- h = perpendicular height of the cone
- s = slant height (length along the curved surface)
- V = volume
- A = total surface area
- A_L = lateral (curved) surface area
- A_B = base area
- π ≈ 3.14159
Example 1: Cone with radius 5 units, height 12 units
Slant Height = √(5² + 12²) = √(25 + 144) = √169 = 13 units
Volume = (1/3) × π × 5² × 12 = (1/3) × π × 300 = 100π ≈ 314.16 units³
Lateral Area = π × 5 × 13 = 65π ≈ 204.20 units²
Base Area = π × 5² = 25π ≈ 78.54 units²
Total Surface Area = 65π + 25π = 90π ≈ 282.74 units²
Example 2: Cone with radius 3 units, height 4 units
Slant Height = √(3² + 4²) = √(9 + 16) = √25 = 5 units
Volume = (1/3) × π × 3² × 4 = (1/3) × π × 36 = 12π ≈ 37.70 units³
Lateral Area = π × 3 × 5 = 15π ≈ 47.12 units²
Base Area = π × 3² = 9π ≈ 28.27 units²
Total Surface Area = 15π + 9π = 24π ≈ 75.40 units²
About Right Circular Cone Calculator
A right circular cone is a 3D shape with a circular base and a curved surface that tapers to a point (apex) directly above the center of the base. This calculator finds all key properties: total surface area (A), volume (V), lateral area (A_L), base area (A_B), and slant height from just the radius and height.
When to Use This Calculator
- Architecture: Calculate volumes and surface areas for cone-shaped roofs, towers, or structures
- Engineering: Design cone-shaped tanks, funnels, or components with specific volume or area requirements
- Manufacturing: Determine material needed for cone-shaped products or packaging
- Mathematics Education: Teach students about 3D geometry and cone properties
- Construction: Estimate materials (paint, metal, fabric) for cone-shaped structures
- Design: Plan cone shapes with specific volume or surface area specifications
Why Use Our Calculator?
- ✅ Complete Properties: Calculates A, V, A_L, A_B, and slant height all at once
- ✅ Instant Results: Get all measurements immediately from just radius and height
- ✅ Step-by-Step Display: See formulas and calculations for each property
- ✅ 100% Accurate: Uses precise mathematical formulas
- ✅ Educational: Helps understand cone geometry relationships
- ✅ Completely Free: No registration required
Understanding Right Circular Cones
A right circular cone consists of:
- Circular Base: A circle with radius r and area πr²
- Curved Surface: A surface that tapers from the base to the apex
- Apex: The point at the top, directly above the center of the base
- Height (h): Perpendicular distance from base to apex
- Slant Height (s): Length along the curved surface from base edge to apex = √(r² + h²)
- Volume: One-third the volume of a cylinder with same base and height = (1/3)πr²h
- Lateral Area: Area of the curved surface only = πrs
- Total Surface Area: Lateral area + base area = πr(s + r)
Real-World Applications
Architecture: A cone-shaped roof has radius 10 m and height 15 m. Volume = (1/3)π × 100 × 15 ≈ 1,570.80 m³ (airspace). Total surface area ≈ 942.48 m² (roofing material needed). Lateral area ≈ 628.32 m² (curved surface only).
Manufacturing: A funnel has radius 5 cm and height 12 cm. Slant height = 13 cm, volume ≈ 314.16 cm³ (capacity). Lateral area ≈ 204.20 cm² (material for the curved surface). Base area ≈ 78.54 cm² (circular opening area).
Packaging: A cone-shaped gift box with radius 8 cm and height 15 cm has volume ≈ 1,005.31 cm³ (storage capacity) and total surface area ≈ 622.04 cm² (wrapping paper needed, including base).
Frequently Asked Questions
What is a right circular cone?
A right circular cone is a 3D shape with a circular base and a curved surface that tapers to an apex (point) directly above the center of the base. "Right" means the apex is perpendicular to the base center, and "circular" means the base is a circle.
What's the difference between height and slant height?
Height (h) is the perpendicular distance from the base to the apex. Slant height (s) is the length along the curved surface from the base edge to the apex. They're related by s = √(r² + h²). Slant height is always greater than or equal to height.
Why is the volume formula (1/3)πr²h?
A cone has exactly one-third the volume of a cylinder with the same base and height. This can be proven using calculus (integration) or geometric methods. The formula V = (1/3)πr²h is a fundamental result in geometry.
What's the difference between lateral area and total surface area?
Lateral area (A_L) includes only the curved surface (πrs). Total surface area (A) includes both the curved surface and the circular base (πrs + πr² = πr(s + r)). If you're covering just the sides, use lateral area. If including the base, use total surface area.
Can I calculate if I only know the slant height?
You need both radius and height (or radius and slant height) to calculate all properties. If you know radius and slant height, find height using h = √(s² - r²), then proceed with the calculator.
How is this different from a cylinder?
A cylinder has parallel circular bases and constant radius. A cone has one circular base and tapers to a point. A cone's volume is (1/3) of a cylinder with the same base and height. A cylinder's lateral area is 2πrh, while a cone's is πrs.