Spherical Cap Volume
Volume of a slice cut from a sphere.
Formula
Variables
Example
A dome shape
How it works
This formula gives the volume of the rounded "cap" formed when a sphere is cut by a plane, leaving a top section with height h and original sphere radius R. It comes from integrating circular cross-sections (or equivalently from the geometry of how the cap tapers), which produces a volume proportional to h^2 and also dependent on how large the original sphere is. The factor (3R - h) captures the fact that a taller cap contains more of the sphere's bulk, but not in direct proportion because the cross-sectional circles shrink as you move away from the center.
When to use it
Use this when a spherical object is partially cut by a flat plane and you need the volume of the remaining top or bottom segment. It appears in geometry problems, manufacturing, astronomy, and any situation involving lenses, tanks, droplets, or rounded containers with a sliced-off top.
Worked example, step by step
Find the volume of a spherical cap from a sphere of radius 10 cm with cap height 4 cm. Step 1: Identify the values: R = 10 cm, h = 4 cm. Step 2: Substitute into V = (πh^2/3)(3R - h). V = (π(4^2)/3)(3(10) - 4) Step 3: Simplify the powers and parentheses. V = (16π/3)(30 - 4) V = (16π/3)(26) Step 4: Multiply. V = 416π/3 cm^3 Step 5: Approximate if needed. V ≈ 435.6 cm^3
Common mistakes to avoid
- Using the sphere’s diameter instead of its radius for R
- Measuring h from the center of the sphere instead of from the cutting plane to the cap tip
- Forgetting that the result is in cubic units, so the radius and height units must match
Did You Know?
The spherical cap formula lets you compute the volume of liquid in a tilted round tank or a dome's interior.
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