Math for the Building Trades — ANEW Pre-Apprenticeship Program
3D Volume & 3D Shapes
Volume is the three-dimensional space inside a solid — how much it holds — measured in cubic units (ft³, in³, m³). If area is the floor, volume is the whole room.
On the job: Concrete, gravel, soil, and water are ordered by volume. Concrete comes by the cubic yard (1 yd³ = 27 ft³). Footings, slabs, tank capacity, a load of fill — all volume. Always pour a little extra.
Why cubed? (think back to Unit 4) In Unit 4, converting area divided by 144 because area has two dimensions: 12² = 144. Volume adds the third dimension — the z-axis — so you go one more power: 12³ = 1,728. That is why 1 ft³ = 1,728 in³. Same pattern you already learned, just one more layer. Open the 3D Grapher to build solids on the X, Y, Z axes.
Why 1 ft³ = 1,728 in³ — step inside the cube
In Unit 4 you saw 1 ft² = 144 in² (12×12). Volume adds the third direction — depth, the z-axis — so each of the three axes needs its own ÷12. Step through to watch the cube fill in, layer by layer.
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The takeawayx, y, z — each direction needs its own ÷12. 12×12×12 = 1,728.
Wrong1,728 in³ ÷ 12 = 144 (only x converted)
Correct1,728 in³ ÷ 1,728 = 1 ft³
See the volume fill — unit cubes taking up the space inside
A 4 × 3 × 2 box fills with 1-unit cubes. Each cube is one cubic unit — watch the guts on the inside fill up. The final count is the volume.
Worked Example — Rectangular Prism
A concrete footing is 8 ft long, 2 ft wide, and 3 ft deep. What is its volume?
Step 1: Write the formula: V = L × W × H
Step 2: Substitute: V = 8 × 2 × 3
Step 3: Multiply: V = 48 ft³
In cubic yards: 48 ÷ 27 = 1.78 yd³ — order 2 yards.
V = 48 ft³
Try It — 3D Shapes
Cube — V = s³
How to solve▶
Rectangular Prism — V = L×W×H
How to solve▶
Cylinder — V = πr²H
How to solve▶
Build a 3D shape
Pick a shape, size it, fill it — drag the shape to spin it
Quick builds
Units
drag to spin
Volume formula reference card
Units 4 & 5 — calculator allowed. Keep this by your work.
Cube
V = s³
Rectangular Prism
V = L × W × H
Cylinder
V = π r² H
Unit conversions1 ft³ = 1,728 in³ (12³). 1 yd³ = 27 ft³ (3³). Area divides by 144 (12²); volume divides by 1,728 (12³) — one more power for the third dimension. π ≈ 3.14159.
On the job: Concrete is ordered by the cubic yard. Take the volume in ft³, divide by 27 to get yd³, and round up — always pour a little extra.
Vocab Key Terms
The words that show up in every volume problem — on the page, on the plans, and on the job. Each card has a picture to lock it in.
Tap a card to expand the example.
Ready to Practice?
The Worksheet has 32 volume problems — cubes, prisms, and cylinders. Each has a labeled 3D drawing, step-by-step how-to, and a graph view.