Cone Volume Calculator
Calculate cone volume, slant height, and surface area: V = (1/3)πr²h
Enter Radius and Height
Slant Height: 7.21 units
Lateral Surface Area: 90.60 square units
Total Surface Area: 140.88 square units
Calculations:
Slant Height: l = √(r² + h²) = √(16 + 36) = √52 = 7.21
Volume: V = (1/3)πr²h = (1/3)π(16)(6) = (1/3)(96π) = 100.53 cubic units
Lateral SA: πrl = π(4)(7.21) = 90.60 square units
Total SA: πr(r + l) = π(4)(4 + 7.21) = π(4)(11.21) = 140.88 square units
Cone Formulas
Volume: V = (1/3)πr²h
Slant Height: l = √(r² + h²)
Lateral Surface Area: LA = πrl
Total Surface Area: SA = πr(r + l)
where: - r = radius of the circular base - h = perpendicular height from base to apex - l = slant height (distance along the side) - π ≈ 3.14159
How to Calculate Cone Volume
- Measure the radius: Find the distance from the center of the base to its edge
- Measure the height: Find the perpendicular distance from the base to the tip (apex)
- Square the radius: Calculate r²
- Multiply by height and π: Calculate πr²h
- Divide by 3: Volume = (1/3) × πr²h
Understanding the Cone
A cone is one-third of a cylinder with the same base and height. This is why the volume formula includes the factor 1/3. The slant height, height, and radius form a right triangle, connected by the Pythagorean theorem: l² = r² + h².
Real-World Applications
Cone calculations are used in: - Engineering: Funnel and hopper design - Architecture: Conical roofs and spires - Food industry: Ice cream cones, party hats - Manufacturing: Tapered components and drill bits - Geology: Volcanic cone measurements - Traffic management: Traffic cone specifications
Common Questions
What is the formula for cone volume?
The cone volume formula is V = (1/3)πr²h, where r is the radius of the base and h is the perpendicular height. A cone has exactly one-third the volume of a cylinder with the same base and height.
How do you find the surface area of a cone?
The total surface area of a cone is SA = πr(r + l), where r is the base radius and l is the slant height. The slant height can be found using l = √(r² + h²).
What is the difference between cone height and slant height?
The height (h) is the perpendicular distance from the base to the apex. The slant height (l) is the distance along the side from the base edge to the apex. They are related by l = √(r² + h²).