SURFACE AREA, VOLUME AND CAPACITY (Cuboid, Cube and Cylinder)
198. Volume, Capacity & Surface Area · The three pillars of 3D measurement
Volume, capacity, and surface area definitions hold for any cuboid dimensions.
Three key 3D measures: Volume = the amount of space a solid occupies (measured in cubic units). Capacity = the volume a hollow container can hold (often in litres). Surface Area = the total area of all the solid's outer faces (measured in square units).
What this lesson covers
Try to break it
Drag L, H, and W to change the cuboid's dimensions. Volume = L × H × W, surface area = 2(LH + HW + WL), and capacity matches the volume when measured in the same units. All three readouts update together. Try to change volume without changing any dimension — impossible.
How you build it
Construct a 3D box.
- Draw the front face — rectangle ABCD (length L by height H), the face closest to you.
- Draw the back face — rectangle EFGH, the same size as ABCD, offset up and to the right to show depth.
- Mark point A — the front bottom-left corner.
- Mark point B — the front bottom-right corner.
- Mark point C — the front top-right corner.
- Mark point D — the front top-left corner.
- Mark point E — the back bottom-left corner.
- Mark point F — the back bottom-right corner.
- Mark point G — the back top-right corner.
- Mark point H — the back top-left corner.
- Connect A to E — the first depth edge.
- Connect B to F.
- Connect C to G.
- Connect D to H to complete the cuboid.
The proof, step by step
Prove that volume, capacity, and surface area each describe a distinct property of the cuboid.
- Volume is the space occupied by a solid body.
- Capacity is the internal volume of a container.
- Surface Area is the sum of areas of all faces.
- Formulas: V = L×H×W, SA = 2(LH + HW + WL).
Worked example
A cuboid has length 10 cm, breadth 8 cm, and height 5 cm. Find its volume.
Volume = L × B × H = 10 × 8 × 5 = 400 cm³. The space occupied by the cuboid is 400 cubic centimeters.
- 400 cm³ — correct
- 300 cm³
- 500 cm³
- 600 cm³