Volume and surface area formulas by solid
The table below summarizes the volume and surface-area formula used for each solid.
| Solid | Volume formula | Surface area formula |
|---|---|---|
| Sphere | V = (4/3)πr³ | SA = 4πr² |
| Cube | V = s³ | SA = 6s² |
| Rectangular box | V = lwh | SA = 2(lw + lh + wh) |
| Cylinder | V = πr²h | SA = 2πr² + 2πrh |
| Cone | V = (1/3)πr²h | SA = πr² + πr·slant |
| Square pyramid | V = (1/3)l²h | SA = l² + 2l·slant |
| Hemisphere | V = (2/3)πr³ | SA = 3πr² |
| Conical frustum | V = (1/3)πh(r₁²+r₁r₂+r₂²) | SA = π(r₁²+r₂²) + π(r₁+r₂)·slant |
- For cone, pyramid and frustum, the surface area formula uses the slant height (or slant edge), which is derived internally from radius/edge and height via the Pythagorean theorem — it is not the same as the vertical height entered.
- "Surface area" here means total surface area, including the base(s); it is not the lateral (side-only) surface area — see the dedicated surface area calculator for that breakdown.
- All formulas assume right (not oblique) solids: the apex or top face sits directly above the center of the base.
What is volume?
Volume is the amount of three-dimensional space enclosed by a solid, measured in cubic units (for example cubic meters or cubic feet). Each regular solid has a closed-form formula relating volume to its defining dimensions, such as a sphere's radius or a box's length, width and height.
Surface area, shown alongside volume for every solid here, is the total area of all exterior faces or curved surfaces. It uses square units and is calculated with a separate formula from volume — the two describe different physical properties and are not interchangeable.
How to use this volume calculator
- Select the solid you want to measure from the solid dropdown.
- Enter the dimensions requested — for example radius for a sphere, or length, width and height for a box.
- For a conical frustum, enter the bottom radius, the top radius (radius2) and the perpendicular height between the two circular faces.
- For a square pyramid, enter the base edge length as "length" and the perpendicular height from base to apex.
- Read the volume and surface area, which update instantly as you change dimensions.
The formula behind each solid's volume
Each solid's volume formula is derived from calculus (integrating cross-sectional area along the solid's height or radius) but reduces to a simple closed-form expression for these standard shapes.
Common mistakes
- Entering diameter instead of radius for a sphere, cylinder, cone or hemisphere, which overstates the volume by a factor of 8.
- Confusing the cube's edge length with a box's independent length, width and height.
- For a frustum, entering the top radius (radius2) larger than the bottom radius without realizing which face is which — the formula is symmetric, but labeling determines orientation.
- Mixing units, such as entering height in feet while radius is in inches, which produces an incorrect result.
- Assuming volume and surface area scale the same way — doubling a sphere's radius multiplies volume by 8 but surface area by only 4.
Preguntas frecuentes
How do you find the volume of a sphere?
A sphere's volume equals (4/3) × π × radius³. A sphere with radius 3 has a volume of (4/3) × π × 27 ≈ 113.0973 cubic units.
What is the volume of a cylinder?
A cylinder's volume equals π × radius² × height. A cylinder with radius 3 and height 5 has a volume of π × 9 × 5 ≈ 141.3717 cubic units.
How do you calculate the volume of a cone?
A cone's volume is one-third of π × radius² × height. A cone with radius 3 and height 5 has a volume of (1/3) × π × 9 × 5 ≈ 47.1239 cubic units — exactly one-third the volume of a cylinder with the same radius and height.
What is the volume of a rectangular box?
A box's volume equals length × width × height. A box measuring 4 × 3 × 5 has a volume of 4 × 3 × 5 = 60 cubic units.
How do you find the volume of a conical frustum?
A frustum's volume equals (1/3) × π × height × (r₁² + r₁r₂ + r₂²), where r₁ and r₂ are the two circular face radii. A frustum with bottom radius 3, top radius 2 and height 5 has a volume of (1/3) × π × 5 × (9 + 6 + 4) ≈ 99.4838 cubic units.
Why does the calculator also show surface area?
Volume and surface area are both standard descriptors of a solid and are frequently needed together — for example volume for capacity and surface area for material or coating requirements — so both are computed from the same entered dimensions.
What is the difference between a sphere and a hemisphere formula?
A hemisphere is exactly half a sphere by volume: V = (2/3)πr³ versus (4/3)πr³ for a full sphere. Its surface area is not simply half the sphere's, because it also includes the flat circular base: SA = 3πr² (2πr² curved + πr² flat), compared with 4πr² for a full sphere.
Referencias
- NIST Guide to the SI — cubic unit conventions.
- Weisstein, Eric W. "Conical Frustum." MathWorld — A Wolfram Web Resource.
- Standard solid-geometry textbook conventions (e.g. Larson, Geometry, Cengage Learning).