Volume of a Spherical Cap: Formula and Examples

A spherical cap is a section of a sphere sliced off by a plane. It is a dome with a circular base. A hemisphere is a special type of cap constructed by cutting a sphere exactly in half. You can calculate the volume of a spherical cap if you know the radius of its base (A) and its height (H). Or you can calculate it from the radius of the larger sphere (R) and the height of the cap.

The three parameters A, H, and R are interrelated by the equivalent expressions

R = (A^2 + H^2)/(2H)
A = sqrt(2RH - H^2)
H = R - sqrt(R^2 - A^2)

This means if you are given any two of the values you can also figure out the remaining value. If you want to compute the volume of a spherical cap, here are two formulas along with several example problems to help you apply the equations in practical settings.

Spherical Cap Volume Formulas

If you are given the height of the cap (H) and the radius of the cap's base (A), then you can figure its volume with the formula

V1 = (π/6) * H * (3*A^2 + H^2)

If you don't know the radius of the cap's base, but you do know the radius of the larger sphere (R), then you can compute its volume with the alternative formula

V2 = (π/3) * H^2 * (3*R - H)

Now, what if you don't know the height of the cap, but you do know both the radius of the base and the radius of the larger sphere. Is there a volume formula in terms of just R and A? Yes! Take the second formula and replace H with the expression R - sqrt(R^2 - A^2), then simplify. This gives you the third volume formula:

V3 = (π/3) * [2*R^3 - (2*R^2 + A^2)*sqrt(R^2 - A^2)]

Example 1

A spherical cap has a circumference of 50.71 cm at its base and a height of 4.94 cm. What is its volume?

To find the radius of the base, divide its circumference by 2π. This gives us A = 50.71/2π = 8.07. With the value H = 4.94, we can use the first volume formula.

V = (π/6)(4.94)(3*8.07^2 - 4.94^2)
= (π/6)(4.94)(170.9711)
= 442.23 cm^3

Example 2

A spherical cap is cut from a sphere with a radius of 70 cm. The resulting cap has height of 36 cm. What is its volume rounded to the nearest whole cubic centimeter?

For this problem, since we are given the values of R and H, we need to use the second volume equation

V = (π/3) * H^2 * (3R - H)

We plug in the two measurements R = 70 and H = 36, which gives us

V = (π/3)(36^2)(3*70 - 36)
= (π/3)(1296)(174)
= 236147 cm^3

Example 3

A spherical cap is cut from a sphere so that the radius of the cap's base is 1/2 the radius of the sphere. If the cap has a volume of 327 cubic inches, what is the height of the cap?

This is a more complicated problem that asks us to find one of the cap's measurements given its volume. Luckily, we have enough information to determine H, as well as R and A. First we note that A = R/2, or equivalently 2A = R. If we use the third volume formula and replace R with 2A, we get

V = (π/3) * [2*R^3 - (2*R^2 + A^2)*sqrt(R^2 - A^2)]

327 = (π/3)[16*A^3 - (8*A^2 + A^2)*sqrt(4*A^2 - A^2)]

= (π/3)[16*A^3 - (9*A^2)*sqrt(3*A^2)]

= (π/3)[16*A^3 - (9*sqrt(3))*A^3]

= (A^3)(π/3)[16 - 9*sqrt(3)]

Now we can solve for A^3 and A:

327/{(π/3)[16 - 9*sqrt(3)]} = A^3

758.7596 = A^3

9.1208 = A

Since we know that A = 9.1208 inches, we also know that R = 2*A = 18.2417 inches. Finally, we use the equation

H = R - sqrt(R^2 - A^2)

to solve for H. This gives us

H = 18.2417 - sqrt(18.2417^2 - 9.1208^2)

= 2.4439 inches.

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Comments 2 comments

slna111 23 months ago

I need to find the height of the spherical cap, given the volume of the cap and the radius of the sphere. (Not the base radius.) What would the formula be for that?

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calculus-geometry 20 months ago from Germany Author

Hi slna,

Solving this spherical cap problem is equivalent to solving a cubic equation in h,

h^3 + ah^2 + b = 0

where a = -3r and b = 3V/pi; r is the given sphere radius and V is the given spherical cap volume. The solution to this equation is a rather complicated expression involving cube roots of square roots of polynomials in a and b, here it is in all its gory detail

h = -a/3 + (1/(3*cbrt(2))[sqrt(108ba^3 + 729b^2) - 2a^3 - 27b]^(1/3) + (a^2)(cbrt(2)/3)[[sqrt(108ba^3 + 729b^2) - 2a^3 - 27b]^(-1/3)

where cbrt is the cube root.

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