Spherical Cap Volume Calculator
Calculate spherical cap volume instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
V = (pi h^2 / 3)(3R - h)
Where V is the volume, R is the sphere radius, and h is the height of the cap. Equivalently, V = (pi h / 6)(3a^2 + h^2) where a is the base radius. The lateral surface area is simply 2 pi R h.
Worked Examples
Example 1: Spherical Cap with Known Radius and Height
Problem:Find the volume and surface area of a spherical cap with sphere radius R = 10 and cap height h = 3.
Solution:Volume = (pi * 3^2 / 3)(3 * 10 - 3)\n= (pi * 9 / 3)(27)\n= 3pi * 27 = 81pi = 254.4690\nBase radius a = sqrt(3(20 - 3)) = sqrt(51) = 7.1414\nLateral area = 2pi * 10 * 3 = 60pi = 188.4956\nBase area = pi * 51 = 160.2212
Result:Volume = 254.4690 | Lateral area = 188.4956 | Base radius = 7.1414
Example 2: Hemisphere Volume Calculation
Problem:Verify the hemisphere volume formula using the spherical cap formula with R = 5, h = 5.
Solution:Cap volume = (pi * 25 / 3)(15 - 5)\n= (25pi/3)(10) = 250pi/3 = 261.7994\nHemisphere formula = (2/3)pi * R^3 = (2/3)pi * 125\n= 250pi/3 = 261.7994\nBoth formulas agree perfectly\nBase radius = sqrt(5 * 5) = 5 (equals R, as expected)
Result:Hemisphere volume = 261.7994 (verified by both formulas)
Frequently Asked Questions
What is a spherical cap and how is its volume calculated?
A spherical cap is the portion of a sphere that is cut off by a plane. Imagine slicing through a sphere with a flat cut: the dome-shaped piece above (or below) the cut is the spherical cap. The volume formula is V = (pi h^2 / 3)(3R - h), where R is the sphere radius and h is the height of the cap (the perpendicular distance from the cutting plane to the top of the cap). An equivalent formula using the base radius a is V = (pi h / 6)(3a^2 + h^2). When h = R, the cap is a hemisphere with volume (2/3) pi R^3. When h = 2R, the cap is the entire sphere with volume (4/3) pi R^3. The formula can be derived by integration, revolving the circular cross-section around the vertical axis.
How do you derive the spherical cap volume formula using calculus?
The volume is derived using the disk method of integration. Place the sphere of radius R centered at the origin. The cap of height h sits between y = R - h and y = R. At height y, the cross-sectional circle has radius r(y) = sqrt(R^2 - y^2). The volume is the integral from (R - h) to R of pi(R^2 - y^2) dy. Evaluating: pi[R^2 y - y^3/3] from R-h to R = pi[(R^3 - R^3/3) - (R^2(R-h) - (R-h)^3/3)]. After algebraic simplification, this yields V = (pi h^2/3)(3R - h). This derivation demonstrates the power of integration for computing volumes of revolution and can be extended to find volumes of spherical segments (caps with both top and bottom cut off) by adjusting the integration limits.
What is the lateral surface area of a spherical cap?
The lateral (curved) surface area of a spherical cap is given by the remarkably simple formula A = 2 pi R h, where R is the sphere radius and h is the cap height. This formula was discovered by Archimedes, who proved that the lateral area depends only on the sphere radius and cap height, not on the position of the cutting plane. This means that any two caps of the same height on the same sphere have equal lateral surface areas, regardless of where they are cut. The total surface area of a cap (including the circular base) is 2 pi R h + pi a^2, where a is the base radius. For a hemisphere (h = R), the lateral area is 2 pi R^2, which equals half the sphere surface area, and the base area is pi R^2, giving a total of 3 pi R^2.
What are real-world applications of spherical cap calculations?
Spherical cap calculations appear in numerous practical fields. In architecture, domes and cupolas are spherical caps, and computing their volume and surface area is essential for material estimation, air volume for HVAC design, and structural analysis. In manufacturing, convex or concave lens surfaces are spherical caps, and their volume determines material requirements. In geography, the area of a polar ice cap can be modeled as a spherical cap on Earth to estimate ice volume. In medicine, the volume of a tumor approximated as a spherical cap helps estimate growth rates. In food science, the volume of liquid in a spherical-bottomed vessel is a spherical cap calculation. In astronomy, the solid angle subtended by a spherical cap determines the fraction of sky observed by a telescope.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy