Volume Calculator — Cube, Sphere, Cylinder & More

3D Shape Volume Guide

What do I need to know about Volume Formulas?

Cube: V = a³. Cuboid: V = l × w × h. Cylinder: V = πr²h. Sphere: V = (4/3)πr³. Hemisphere: V = (2/3)πr³. Cone: V = (1/3)πr²h. Square pyramid: V = (1/3) × base area × h = (1/3)a²h. Triangular prism: V = (1/2) × base × height × length = (1/2)bhl. Torus: V = 2π²Rr² (R = major radius from centre to tube centre, r = tube radius). The factor of 1/3 appears in both cone and pyramid — they share the principle that a cone/pyramid is 1/3 of the volume of the cylinder/prism with the same base and height.

What's the key thing to understand about Surface Area Formulas?

Cube: SA = 6a². Cuboid: SA = 2(lw + lh + wh). Cylinder: SA = 2πr² + 2πrh = 2πr(r+h). Sphere: SA = 4πr². Hemisphere: SA = 3πr² (2πr² curved + πr² flat). Cone: SA = πrl + πr² = πr(l+r) where l = √(r²+h²) is the slant height. Square pyramid: SA = a² + 2al where l = slant height of triangular face = √(h²+(a/2)²). The slant height is always measured along the face, not the vertical height — a common error.

What do I need to know about Units and Conversions?

Volume units: 1 m³ = 1,000,000 cm³ = 1,000 L. 1 cm³ = 1 mL. 1 L = 1,000 cm³. 1 m³ = 1,000 L. Surface area units are always the square of the linear unit. When dimensions are given in cm: volume in cm³, surface area in cm². Converting: if sides change by a factor of k, volume changes by k³ and surface area by k². Example: doubling all sides of a sphere: radius × 2. Surface area × 4. Volume × 8. This 'scaling law' explains why large animals have lower surface-area-to-volume ratios — relevant to heat loss, since smaller animals lose heat faster relative to their body mass and need proportionally more energy to stay warm.

What's the key thing to understand about Real-World Applications?

Cylinders: water tanks, pipes, silos, cans. A cylinder with r=h is the optimal shape for a can (minimises material per unit volume). Spheres: storage tanks for liquids and gases (lowest surface area per volume). Planets and droplets are spherical for the same reason. Cones: ice cream cones, funnels, dunce caps (sorry). Pyramids: structurally very stable — the Great Pyramid of Giza has withstood 4,500 years. Prisms: structural beams (I-beams have cross-section designed for bending resistance). Prisms with a consistent cross-section, like extruded aluminium window frames, are manufactured this way specifically because the volume and material cost scale predictably with length.

What should I know about Volume by Integration (Cross-Sections)?

For an irregular solid, volume can be found by integrating the cross-sectional area A(x) along its length: V = ∫A(x)dx between the solid's bounds. This is how the standard formulas on this page were themselves derived — a cone's cross-section at height h from the apex is a circle whose radius scales linearly with h, so integrating πr²(h) from 0 to the cone's full height produces exactly (1/3)πr²h, confirming the factor of ⅓ that distinguishes a cone from a cylinder of the same base and height. This technique, called the 'disk method' when cross-sections are circles, is how volumes of revolution (spinning a 2D curve around an axis to generate a 3D solid) are calculated in calculus, and it generalises the standard geometric formulas to any shape whose cross-sectional area can be expressed as a function of position.

What do I need to know about Converting Between Volume Units?

Volume units scale as the cube of the linear conversion factor, which trips up far more people than linear or area conversions. 1 metre = 100 cm, but 1 cubic metre = 100³ = 1,000,000 cubic centimetres, not 100. Similarly, 1 cubic metre = 1000 litres exactly (since 1 litre is defined as 1000 cm³, and 1 m³ = 1,000,000 cm³ = 1000 litres). Converting cubic feet to cubic metres: multiply by 0.0283 (since 1 foot = 0.3048m, and 0.3048³ ≈ 0.0283). A common costly mistake in concrete or materials ordering is converting a volume using the linear conversion factor instead of its cube — for example, treating '3 times bigger in each dimension' as '3 times the volume' instead of the correct 27 times (3³), which can lead to drastically under-ordering materials for a scaled-up design.

What's the key thing to understand about Volume Scaling with Linear Dimensions?

When every linear dimension of a 3D shape is scaled by a factor k, its volume scales by k³, not by k — a relationship that catches people off guard surprisingly often. Doubling a cube's side length (k = 2) doesn't double its volume; it multiplies it by 2³ = 8. This is why a "jumbo" size of a product that looks only modestly larger than the "regular" size can actually hold several times the volume, and why small changes in a container's dimensions have an outsized effect on its capacity. The same k³ scaling applies to any 3D shape, not just cubes, since every volume formula is built from three length measurements multiplied together — which is also why surface area, built from only two length measurements, scales by k² instead, a mismatch that matters enormously in biology (why larger animals need proportionally thicker legs) and engineering (why scaled-up models don't behave structurally like the full-size original).

What should I know about Surface-Area-to-Volume Ratio with Scale?

What should I know about a Worked Example: A Composite Solid?

As a worked example, consider a test tube modelled as a cylinder of radius 1.5cm and height 10cm, capped at the bottom with a hemisphere of the same radius. Cylinder volume: V=πr²h=π×2.25×10=22.5π≈70.69 cm³. Hemisphere volume: V=(2/3)πr³=(2/3)π×3.375=2.25π≈7.07 cm³. Total volume: ≈77.75 cm³. For surface area, only exposed curved surfaces count: curved cylinder surface 2πrh=30π≈94.25 cm²; curved hemisphere surface 2πr²=4.5π≈14.14 cm²; open top circle πr²≈7.07 cm². Total exposed glass surface ≈108.4 cm². This 'build from simple pieces' approach — add volumes, but carefully include only genuinely exposed surfaces — is the standard method for any irregular composite object.

What should I know about the Volume of a Frustum?

A frustum is a cone or pyramid with its tip sliced off by a plane parallel to the base. For a conical frustum with top radius r, bottom radius R, and height h: V=(πh/3)×(R²+Rr+r²) — a formula that correctly reduces to the standard cone volume when r=0 and to a cylinder's volume when r=R. Worked example: a bucket shaped as a frustum with bottom radius 15cm, top radius 20cm, and height 30cm holds V=(π×30/3)×(400+300+225)=10π×925=9,250π≈29,060 cm³, about 29 litres. Frustums describe buckets, lampshades, plant pots, and the sides of truncated pyramids — any tapered container that doesn't come to a complete point.

What should I know about the Volume of an Ellipsoid?

An ellipsoid — a sphere stretched unevenly along up to three different axes — has volume V = (4/3)πabc, where a, b, and c are its three semi-axis lengths. Setting a=b=c=r recovers the familiar sphere formula, (4/3)πr³, confirming the ellipsoid formula as the sphere's natural generalisation. Worked example: a rugby ball modelled as an ellipsoid with semi-axes 15cm, 9cm, and 9cm has volume V = (4/3)π×15×9×9 = (4/3)π×1,215 = 1,620π ≈ 5,089 cm³ (about 5.1 litres). This formula is used for modelling planets (which bulge slightly at the equator due to rotation, making them very slightly ellipsoidal rather than perfectly spherical), eggs, and any roughly egg- or rugby-ball-shaped object whose three perpendicular dimensions genuinely differ.

What should I know about the Surface Area of an Ellipsoid?

Unlike its volume, an ellipsoid's exact surface area has no simple closed-form formula in general (it requires elliptic integrals), but a widely used and reasonably accurate approximation (Knud Thomsen's formula) is SA ≈ 4π × [((a^p b^p + a^p c^p + b^p c^p)/3)]^(1/p), with p≈1.6075 giving less than 1.1% error for typical cases. This is a useful reminder that not every geometric shape has a tidy formula the way a sphere or cuboid does — some genuinely require either an approximation, like this one, or numerical integration to evaluate precisely, which is exactly the kind of calculation that computer software now handles routinely but that would have been extremely laborious by hand before computers were available.

What should I know about Why Pi Appears in Curved-Shape Volumes?

Pi appears in every volume formula involving a circular cross-section (cylinder, cone, sphere) because it is fundamentally the ratio relating any circle's circumference and area to its radius — any shape built from circular slices inherits a factor of pi for exactly this reason, while shapes built from flat polygonal cross-sections (cubes, prisms, pyramids) never need it at all.