Geometry · 3D Shapes
Volume
Calculator
Calculate volume and surface area for 10 three-dimensional shapes. Supports multiple units with conversion, shape illustrations, and step-by-step formulas.
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Volume
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Surface Area
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Shape
🔮 Sphere
V = (4/3) × π × r³
SA = 4 × π × r²
Volume in cm³, SA in cm²
Volume
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units³
Surface Area
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units²
Volume
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Surface Area
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Lateral SA
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📋 Step-by-Step Solution
Volume Formulas for 3D Shapes
Volume measures the amount of 3D space a solid occupies, expressed in cubic units. Surface area measures the total area of all faces or curved surfaces, in square units.
All Volume Formulas
Sphere: V = (4/3)πr³ SA = 4πr²
Cube: V = a³ SA = 6a²
Rect. Box: V = l×w×h SA = 2(lw+lh+wh)
Cylinder: V = πr²h SA = 2πr² + 2πrh
Cone: V = (1/3)πr²h SA = πr² + πrl (l=slant height)
Pyramid: V = (1/3)×a²×h SA = a² + 2a×l
Tri. Prism: V = (1/2)×b×h_t×L SA = bL + perimeter×L + 2×(triangle area)
Capsule: V = πr²(4r/3+h) SA = 2πr(2r+h)
Ellipsoid: V = (4/3)πabc (approx SA: 4π×((a^p×b^p+a^p×c^p+b^p×c^p)/3)^(1/p), p≈1.6075)
Torus: V = 2π²Rr² SA = 4π²Rr
Why does a cone have 1/3 the volume of a cylinder?
A cone with the same base and height as a cylinder fills exactly 1/3 of the cylinder's volume. This can be proven geometrically by decomposing a cube into three equal pyramids. You can verify it practically: fill a cone-shaped container with water, pour it into a cylinder of the same base and height — it takes exactly 3 fills to fill the cylinder. The same relationship holds for pyramids vs prisms.
What is the most efficient shape by volume-to-surface-area ratio?
A sphere has the highest volume-to-surface-area ratio of any 3D shape — it encloses the most volume for the least surface area. This is why bubbles and drops are spherical (minimizing surface tension), and why cells tend toward spherical shapes. For a given volume V, the sphere has SA = (36π)^(1/3) × V^(2/3) — less surface area than any other shape with the same volume.
What is the difference between lateral surface area and total surface area?
Lateral surface area (LSA) is the area of all faces except the base(s) — just the "sides." Total surface area (TSA) includes the base(s). For a cylinder: LSA = 2πrh (just the curved side), TSA = 2πrh + 2πr² (sides + top + bottom). For a cone: LSA = πrl (just the slanted side), TSA = πrl + πr² (side + base). Lateral SA is useful for painting just the sides of a container.
How does doubling the radius affect the volume of a sphere?
Volume = (4/3)πr³. Doubling r gives (4/3)π(2r)³ = (4/3)π×8r³ = 8 × original volume. So doubling the radius multiplies the volume by 8 (2³). Tripling the radius multiplies it by 27 (3³). This cubic relationship means small changes in radius create large volume changes — a 10% increase in radius gives 1.1³ ≈ 1.33 (33% more volume). This is why the Earth's volume is 49× that of Mars despite having only about 2× the diameter.