Solids are where geometry meets everyday quantities: liters in a tank, cubic yards of concrete, square feet of sheet metal. This page brings the standard volume and surface area formulas for 13 solids together. Choose a shape in the tool and you get its formula table, every formula worked out with your own numbers, the related measures like slant height and diagonals, and a labeled drawing.
How to use the 3D formula sheet
- Pick a Shape from the list.
- Enter the dimensions that appear. Each field is labeled with the letter used in the formulas, such as r for radius or h for height.
- Choose the Units.
- The volume is the headline, followed by the surface area, its parts and the extra measures. Formulas with your numbers shows each calculation step by step, and the table under the diagram lists the general formulas for that shape.
Volume and surface area formulas
| Solid | Volume | Surface area | Other |
|---|---|---|---|
| Cube (a) | a³ | 6a² | space diagonal a√3 |
| Rectangular prism (l, w, h) | lwh | 2(lw + lh + wh) | diagonal √(l² + w² + h²) |
| Sphere (r) | ⁴⁄₃πr³ | 4πr² | great circle 2πr |
| Hemisphere (r) | ⅔πr³ | 3πr² | curved part 2πr² |
| Cylinder (r, h) | πr²h | 2πr(r + h) | side 2πrh |
| Cone (r, h) | ⅓πr²h | πr(r + s) | s = √(r² + h²) |
| Conical frustum (R, r, h) | ⅓πh(R² + Rr + r²) | π(R + r)s + πR² + πr² | s = √((R − r)² + h²) |
| Capsule (r, a) | πr²(⁴⁄₃r + a) | 2πr(2r + a) | length a + 2r |
| Square pyramid (a, h) | ⅓a²h | a² + 2as | s = √(h² + a² ⁄ 4), edge √(h² + a² ⁄ 2) |
| Pyramid frustum (A₁, A₂, h) | ⅓h(A₁ + A₂ + √(A₁A₂)) | A₁ + A₂ + n(a + b)s ⁄ 2 | s from the apothems |
| Triangular prism (a, b, c, L) | BL | 2B + (a + b + c)L | B by Heron’s formula |
| Tube (R, r, L) | π(R² − r²)L | 2π(R + r)L + 2π(R² − r²) | hollow volume πr²L |
| Ellipsoid (a, b, c) | ⁴⁄₃πabc | ≈ Thomsen’s formula, p = 1.6075 | within about 1.1% |
Worked example: a cone
A paper cone has a base radius of 3 cm and a height of 4 cm (the tool’s starting example).
- Volume: ⅓ × π × 3² × 4 = 12π ≈ 37.70 cm³.
- Slant height: √(3² + 4²) = 5 cm, the length of the paper from rim to tip.
- Side area: π × 3 × 5 = 15π ≈ 47.12 cm²; base: π × 3² = 9π ≈ 28.27 cm².
- Total surface: 24π ≈ 75.40 cm². A cone open at the base needs only the 47.12 cm² of paper.
Patterns behind the formulas
Prisms and cylinders: base times height
A box, a cylinder, a triangular prism and a tube all keep the same cross-section from end to end, so each volume is the base area times the length. Knowing the area of a circle, triangle or ring is enough to rebuild all of them.
Pyramids and cones: one-third of that
Any solid that tapers evenly to a point holds one-third of its matching prism or cylinder: ⅓a²h for a square pyramid, ⅓πr²h for a cone. A frustum is the big pyramid or cone minus the small one cut from its top, which produces the three-term sum ⅓h(A₁ + A₂ + √(A₁A₂)).
Spheres: Archimedes’ two-thirds
Archimedes proved that a sphere has two-thirds the volume and two-thirds the total surface of the cylinder that just encloses it. From the cylinder πr² × 2r = 2πr³, two-thirds gives ⁴⁄₃πr³. A capsule is a cylinder plus one whole sphere, which is why its formulas simply add the two.
Cavalieri’s principle
If two solids have equal cross-sectional areas at every height, they have equal volumes. A slanted stack of coins has the same volume as a straight stack, so the volume formulas hold for oblique prisms and cones when h is the perpendicular height.
Scaling
Multiplying every length by k multiplies surface area by k² and volume by k³. A model built at 1:10 scale needs one-hundredth of the paint and holds one-thousandth of the volume of the full-size object.
Getting units right
Volume comes out in cubic units and area in square units of whatever length you enter. Convert before you calculate: 1 ft³ = 1,728 in³ = 7.4805 US gallons, 1 yd³ = 27 ft³, and 1 m³ = 1,000 liters.
For flat shapes, see the 2D shape formulas. To solve a single solid backward, for example a cylinder’s height from its volume, use a dedicated tool such as the cone calculator or the volume calculator.
Frequently asked questions
Why is the volume of a cone one-third of a cylinder?
A cone with the same base and height as a cylinder holds exactly one-third as much. Slicing both into thin disks shows the cone's disks shrink steadily to a point; adding them up (a calculus sum of r² terms) gives one-third. The same rule links every pyramid to its prism.
What is the difference between height and slant height?
Height is measured straight up from the base to the top, at a right angle. Slant height runs along the sloping surface of a cone or pyramid face, so it is longer. Volume formulas use the height; side-area formulas use the slant height.
What happens to volume and surface area when a shape is scaled up?
If every length is multiplied by k, surface area is multiplied by k² and volume by k³. Doubling a ball's diameter gives four times the surface and eight times the volume.
Do these formulas work for slanted (oblique) prisms and cylinders?
The volume formulas do, as long as you use the perpendicular height. That is Cavalieri's principle: equal cross-sections at every level mean equal volume. The surface area formulas on this page assume right (upright) solids.
Which shapes are on the formula sheet?
Cube, rectangular prism, sphere, hemisphere, cylinder, cone, conical frustum, capsule, square pyramid, pyramid frustum with any regular base, triangular prism, tube and ellipsoid.