Volume Calculator
Calculate the volume and surface area of 8 three-dimensional shapes. Select a shape, enter dimensions, and get volume, surface area, and key measurements instantly.
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How It Works
All three dimensions are equal. Volume = s³. Surface area = 6s². The space diagonal passes through the center: d = s√3.
V = s³, SA = 6s²s=5 → V=125, SA=150Volume = length × width × height. Surface area = 2(lw + lh + wh). The space diagonal = √(l²+w²+h²).
V = l×w×h, SA = 2(lw+lh+wh)6×4×3 → V=72, SA=108Volume = (4/3)πr³. Surface area = 4πr². A sphere encloses the maximum volume for a given surface area — the most efficient 3D shape.
V = (4/3)πr³, SA = 4πr²r=5 → V=523.6, SA=314.2Volume = πr²h. Surface area (closed) = 2πr(r+h). The base area is πr². Think of it as a stack of circles.
V = πr²h, SA = 2πr(r+h)r=3, h=8 → V=226.2, SA=207.3Volume = (1/3)πr²h — one-third of a cylinder with the same base and height. Slant height l = √(r²+h²).
V = (1/3)πr²h, SA = πr(r+l)r=4, h=9 → V=150.8Volume = (1/3) × base area × height. For a square base: V = (1/3)s²h. A pyramid is 1/3 of a prism with the same base and height.
V = (1/3) × B × hbase=6, h=9 → V=108Quick Reference
Common examples — verify instantly above.
Cube
side=5
V=125
Box
6×4×3
V=72
Sphere
r=5
V=523.6
Cylinder
r=3, h=8
V=226.2
Cone
r=4, h=9
V=150.8
Pyramid
base=6, h=9
V=108
SA Cube
side=4
SA=96
SA Sphere
r=5
SA=314.2
Tips & Shortcuts
Volume scales with the CUBE of dimensions. Doubling all dimensions multiplies volume by 8.
A cone holds exactly 1/3 the volume of a cylinder with the same base and height.
A sphere has the smallest surface area for a given volume — this is why soap bubbles are spherical.
When measuring radius from a diameter, divide by 2 first. Using diameter directly in sphere formula gives 8× the correct volume.
Surface area of a cylinder: 2πr² (two circles) + 2πrh (lateral rectangle rolled around). Remember both caps.
For the slant height of a cone: l = √(r²+h²). This is needed for lateral surface area, not volume.
Common Mistakes
Using diameter instead of radius in sphere and cylinder formulas
All volume formulas use radius (r). Diameter = 2r. Divide diameter by 2 first.
Forgetting the 1/3 factor for cone and pyramid volumes
Cone and pyramid volumes are always 1/3 × base area × height. Without the 1/3, the result is the volume of the enclosing prism/cylinder.
Confusing surface area and volume units
Volume is in cubic units (m³). Surface area is in square units (m²). They cannot be compared or added.
Computing lateral surface area instead of total surface area
Total surface area includes the base(s). Lateral area is just the sides. Know which is asked before calculating.
Squaring radius instead of cubing for sphere volume
Sphere volume uses r³ (cubed). Sphere surface area uses r² (squared). These are different formulas for different properties.
Using slant height instead of vertical height in cone volume
Volume uses the vertical height (h), not the slant height (l). Slant height is used only for lateral surface area.
Frequently Asked Questions
Volume is the amount of three-dimensional space inside a shape, measured in cubic units (cm³, m³, ft³, etc.).
V = (4/3)πr³. A sphere's volume scales with the cube of its radius.
Volume measures space inside a 3D object. Surface area measures the total area of all outer faces.
V = πr²h — the area of the circular base multiplied by the height.
A cone with the same base and height as a cylinder has exactly 1/3 its volume: V = (1/3)πr²h.
A frustum is a cone with its top cut off. Its volume is (πh/3)(R² + Rr + r²) where R and r are the radii of the two circular ends.
A cone's volume is exactly 1/3 of a cylinder with the same base radius and height: V_cone = (1/3)πr²h vs V_cylinder = πr²h. It takes exactly three cone-fills to fill the corresponding cylinder — provable by Cavalieri's principle or calculus integration. The same 1/3 relationship holds for a pyramid vs a rectangular prism: V_pyramid = (1/3) × base area × height.
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