Volume formulas, in 3D
Drag any 3D shape with your finger to rotate it, move the sliders to resize, and watch the volume formula update live. Built for Year 4–6 maths and scholarship preparation — cube, rectangular prism, triangular prism, cylinder, cone, square pyramid, plus composite shapes like an L-prism, a house-prism, a staircase, and a lighthouse.
Calculate the volume of a cube
V = 125 cm³
Tip: drag the 3D shape to rotate it, then move any slider and watch the formula update.
How to find the volume of any 3D shape
Volume is the amount of space a 3D shape takes up — measured in cubic units like cm³ or m³. The good news: most of the formulas you'll meet at primary school reduce to the same idea — find a base area, then multiply by the height (or by a third of the height, for pointy shapes).
The cross-section trick
Any shape with a constant cross-section — a prism, a cylinder — has volume equal to its cross-section area multiplied by its depth (or height). That's why six of the formulas below look similar: each one is just (base area) × depth, with a different way of computing the base area. Pyramids and cones tip forward to a point, so they take a third of that.
Volume of a cube
A cube has six identical square faces, so every dimension is the same: side × side × side. That's where 'cubed' comes from in maths.
Example: A 5 cm cube has a volume of 5 × 5 × 5 = 125 cm³.
Doubling the side of a cube doesn't double the volume — it multiplies it by 8. Scholarship questions love testing this 'scale factor' intuition.
Volume of a rectangular prism
A rectangular prism (also called a cuboid) is the everyday box shape — a shoebox, a swimming pool, a brick. Its volume is width × height × depth, with all three measurements in the same unit.
Example: A 8 m × 4 m × 5 m swimming pool has a volume of 8 × 4 × 5 = 160 m³.
The order of multiplication doesn't matter — w × h × d gives the same answer as d × w × h. Pick whichever pairing makes the arithmetic friendliest.
Volume of a triangular prism
A triangular prism looks like a Toblerone bar or a tent. Its cross-section is a triangle, so the area of that face is ½ × base × triangle-height. Multiply by the depth (length) of the prism for the volume.
Example: A tent with a 6 m base, 5 m apex height, and 8 m depth has a volume of ½ × 6 × 5 × 8 = 120 m³.
The 'height' in this formula is the perpendicular height of the triangular face — not the slanted edge. Mark the right-angle in the diagram and measure straight up from the base.
Volume of a cylinder
A cylinder is a circular prism — its cross-section is a circle (π × radius²). Multiply that by the height for the full volume. Tanks, drink cans, mugs and pipes are all cylinders.
Example: A water tank with a 3 m radius and 8 m height has a volume of π × 3² × 8 = 72π ≈ 226.19 m³.
Always check whether the question gives radius or diameter. Diameter is twice the radius, so squaring it gives four times the correct answer — a classic mistake on scholarship papers.
Volume of a cone
A cone tapers from a circular base up to a point. Its volume is exactly one-third of a cylinder with the same radius and height — see the showcase below for a visual proof.
Example: An ice-cream cone with a 4 cm radius and 9 cm height has a volume of ⅓ × π × 4² × 9 = 48π ≈ 150.80 cm³.
The ⅓ shows up because three cones of the same radius and height fit exactly inside the matching cylinder. Forgetting that factor is the single most common cone-volume mistake.
Volume of a square pyramid
A pyramid tapers from a flat base up to a point, just like a cone — so it also uses a ⅓ factor. The base of a square pyramid is a square, so the base area is side × side.
Example: A pyramid with a 6 m square base and 8 m height has a volume of ⅓ × 6² × 8 = 96 m³.
A square pyramid with the same base and height as a cube takes up exactly one third of the cube's volume — for the same reason cones are ⅓ of cylinders.
Volume sits in the measurement strand of the Australian Curriculum for Years 4–6. Practice with real 3D models — rotating, slicing, comparing — is the fastest way to build the spatial sense scholarship exams test for.
Three cones really do fill one cylinder
If a cone and a cylinder share the same radius and height, three full cones tip exactly into one cylinder. Tap Pour to see it happen.
0 of 3 cones poured
Slice it anywhere — same shape
Drag the slider to slice the shape at any height. The cross-section on the right stays the same — that's the defining property of a prism (or cylinder), and it's what makes the formula 'cross-section × depth' work for so many shapes.
Volume = (cross-section area) × depth
If every horizontal slice through a shape looks the same, the volume is just that shape's area, multiplied by how deep the shape goes.
Worked example — find the volume of a swimming pool
A rectangular swimming pool is 8 m long, 5 m wide and 4 m deep. Find its volume in cubic metres.
Step 1 — identify the shape
A swimming pool with rectangular walls and a flat base — that's a rectangular prism.
Step 2 — pick the formula
Rectangular-prism volume is V = width × height × depth, with all three sides in the same unit.
Step 3 — find the dimensions
From the question: width = 8 m, depth = 5 m, height = 4 m.
Step 4 — substitute and calculate
V = 8 × 4 × 5 = 160.
Step 5 — label the units
Lengths were in metres, so the volume is in cubic metres. The pool holds 160 m³ (which is 160,000 litres of water).
Three mistakes kids make (and how to fix them)
Forgetting to cube the unit
Volume is always measured in cubic units — cm³, m³, mm³ — never plain cm or m. The exponent is part of the answer. Scholarship markers deduct marks for missing it, even when the number is right.
Using diameter instead of radius
Cylinder and cone volumes use π × r², not π × d². If the question gives the diameter, halve it before squaring — squaring the diameter gives four times the correct answer.
Forgetting the ⅓ on cones and pyramids
Cones are ⅓ of a cylinder with the same radius and height. Square pyramids are ⅓ of a box with the same base and height. Skip the ⅓ and the answer is three times too big.
Words to know
These are the words your child will hear and need to use confidently when answering volume questions.
- Volume
- The amount of space a 3D shape takes up, measured in cubic units like cm³ or m³.
- Capacity
- The amount a container can hold, usually measured in millilitres or litres. 1 cm³ = 1 mL, so 1,000 cm³ = 1 L.
- Cube
- A 3D shape with six identical square faces. Every edge is the same length.
- Prism
- A 3D shape with two identical end faces (the 'cross-section') joined by rectangular sides. A box is a rectangular prism; a Toblerone is a triangular prism.
- Pyramid
- A 3D shape with a flat base and triangular sides that meet at a single point (the apex).
- Cylinder
- A 3D shape with two identical circular ends and a curved surface joining them — like a tin can.
- Cone
- A 3D shape with a circular base that tapers up to a single point — like an ice-cream cone.
- Cross-section
- The 2D shape you'd see if you cut a 3D shape straight across. Prisms and cylinders have the same cross-section all the way through.
- Radius
- The distance from the centre of a circle to its edge — half the diameter.
- Height
- The straight-up distance from the base to the top of a 3D shape. For cones and pyramids, this is the perpendicular height — straight up to the apex, not along the slanted edge.
- Cubic unit
- A unit of volume. 1 cm³ is the space taken up by a 1 × 1 × 1 cm cube. 1,000 cm³ fit inside a 10 cm cube.
- Pi (π)
- The number 3.14159… It shows up in every circle, cylinder, and cone formula because the area of a circle is π × radius².
Volume questions parents ask
What year level does my child need to know volume formulas?
Cubes and rectangular prisms appear in Year 4; cylinders, pyramids and cones come in Years 5–6. Scholarship exams (ACER, Edutest, AAS) sat in Year 6 routinely test volume of all six shapes plus composite shapes.
Why does the cone formula have a ⅓?
Three cones of the same radius and height fit exactly inside one cylinder. The showcase above lets your child see this happen — once they've watched it, the ⅓ stops feeling arbitrary.
What's the difference between volume and capacity?
Volume is the geometric space a shape takes up (in cm³, m³). Capacity is how much liquid a container holds (in mL, L). They're closely linked: 1 cm³ = 1 mL, so a 1,000 cm³ container holds exactly 1 litre.
Do I need to memorise the cone and pyramid formulas separately?
Not really. Both are ⅓ × base area × height. The only difference is the base shape: for a cone, base area is π × r² (a circle); for a square pyramid, it's side² (a square). One pattern, two bases.
How do I find the volume of a composite shape like an L-shaped pool?
Split the cross-section into rectangles you can compute separately, add or subtract them, then multiply by the depth. The L-prism in the explore tool above does exactly this — drag the cut sliders to see how the cross-section area changes.
Why are some answers given 'in terms of π'?
Because π is irrational, leaving it in the answer (like 72π m³) is exact. Multiplying it out (72 × 3.14159… ≈ 226.19) introduces rounding. Scholarship questions often ask for both forms — the exact π-version first, then the decimal approximation.
Is the volume tool above free to use?
Yes — it's free, no signup required. PrepHQ also runs full practice tests (with adaptive question generation) for families preparing for ACER, Edutest, AAS and ACER HAST scholarship exams.
Which exams test volume?
Volume appears across ACER (scholarship and ACER HAST), Edutest, AAS, and Victorian Curriculum Year 5–6 maths. Knowing all six shape families and the cross-section trick covers every primary-level volume question.