What Is the Paraboloid Volume and Surface Area Calculator?
A circular paraboloid is a quadric surface of revolution formed by rotating a parabola about its central axis of symmetry.
Parabolic reflectors exhibit the optical property that parallel incoming rays reflect directly into a single focal point.
How Does the Paraboloid Volume and Surface Area Calculator Work?
Volume is evaluated using integration: V = ∫ π·(r_z)² dz = ½·π·r²·h.
The curved lateral surface area uses the definite integral of surface revolution: A_lat = (π·r / (6·h²)) · ((r² + 4·h²)^(3/2) - r³).
Paraboloid Volume and Surface Area Calculator Formula & Variables
The core mathematical equation utilized by this calculator is expressed as:
The volume of a circular paraboloid of revolution is exactly one-half the volume of a cylinder with identical base radius and height.
How to Use the Paraboloid Volume and Surface Area Calculator
- Enter the opening radius r at the widest circular aperture.
- Enter the height/depth h from base rim to vertex.
- Review enclosed volume, lateral curved surface area, and flat circular base area.
Step-by-Step Example Calculation
Parabolic Dish (r = 4, h = 6)
Input Values:
Understanding Your Result
The volume of a paraboloid is exactly 50% that of a cylinder with the same radius and height.
In comparison, a cone with identical dimensions occupies 33.3% of the circumscribing cylinder.
Factors That Affect the Result
- Focal Length: Related by f = r² / (4·h). Shallow dishes have longer focal lengths than deep dishes.
- Aspect Ratio: Depth-to-diameter ratio governs optical curvature and material surface consumption.
When Should You Use This Calculator?
- Calculating fluid volume in spinning liquid-mirror telescopes and centrifuge bowls.
- Estimating metallic sheeting and fiberglass needed to construct satellite dishes and radar antennas.
Assumptions & Limitations
- Applies strictly to circular paraboloids of revolution (elliptic paraboloids have differing axes).
Frequently Asked Questions
Calculation Accuracy & Reference Note
Exact analytical integration formulation of quadric revolution surface areas.