Ellipsoid Calculator (Volume & Surface Area)
Ellipsoid Calculator (Volume & Surface Area)
An ellipsoid is a sphere stretched or squashed independently along three perpendicular axes - a rugby ball, an egg, and (very roughly) Earth's own shape are everyday examples. Enter the three semi-axis lengths a, b, and c (half the length of the ellipsoid along each axis; in whatever unit you like), and this calculator finds the volume exactly and the surface area using the widely-used Thomsen approximation, along with a diagram and the full step-by-step working.
- V = 4/3πabc, a direct generalization of the sphere formula (which is the special case a=b=c=r) - semi-axes 5, 3, and 2 give a volume of 4/3 × π × 5 × 3 × 2 ≈ 125.66 cubic units.
- Unlike volume, there is no simple exact formula for an ellipsoid's surface area - this calculator uses the Thomsen approximation, SA ≈ 4π × ((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3)^(1/p) with p ≈ 1.6075, which is accurate to within about 1.1% for all ellipsoid shapes.
- Setting all three semi-axes equal (a=b=c) reduces the formulas back to a sphere's - a useful sanity check, since 4/3πabc becomes 4/3πr³ and the Thomsen approximation becomes exactly 4πr².
How do I find the volume of an ellipsoid?
Use V = 4/3πabc, where a, b, and c are the three semi-axis lengths. For semi-axes 5, 3, 2: V = 4/3 × π × 5 × 3 × 2 ≈ 125.66.
Why is the surface area an approximation instead of an exact formula?
An ellipsoid's exact surface area requires elliptic integrals with no simple closed form. The Thomsen approximation used here is accurate to within about 1.1% in the worst case, which is more than precise enough for practical use.
Ellipsoid Calculator (Volume & Surface Area)
An ellipsoid is a sphere stretched or squashed independently along three perpendicular axes - a rugby ball, an egg, and (very roughly) Earth's own shape are everyday examples. Enter the three semi-axis lengths a, b, and c (half the length of the ellipsoid along each axis; in whatever unit you like), and this calculator finds the volume exactly and the surface area using the widely-used Thomsen approximation, along with a diagram and the full step-by-step working.
- V = 4/3πabc, a direct generalization of the sphere formula (which is the special case a=b=c=r) - semi-axes 5, 3, and 2 give a volume of 4/3 × π × 5 × 3 × 2 ≈ 125.66 cubic units.
- Unlike volume, there is no simple exact formula for an ellipsoid's surface area - this calculator uses the Thomsen approximation, SA ≈ 4π × ((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3)^(1/p) with p ≈ 1.6075, which is accurate to within about 1.1% for all ellipsoid shapes.
- Setting all three semi-axes equal (a=b=c) reduces the formulas back to a sphere's - a useful sanity check, since 4/3πabc becomes 4/3πr³ and the Thomsen approximation becomes exactly 4πr².
How do I find the volume of an ellipsoid?
Use V = 4/3πabc, where a, b, and c are the three semi-axis lengths. For semi-axes 5, 3, 2: V = 4/3 × π × 5 × 3 × 2 ≈ 125.66.
Why is the surface area an approximation instead of an exact formula?
An ellipsoid's exact surface area requires elliptic integrals with no simple closed form. The Thomsen approximation used here is accurate to within about 1.1% in the worst case, which is more than precise enough for practical use.