SURFACE AREA, VOLUME AND CAPACITY (Cuboid, Cube and Cylinder)
201. The Box's Hidden Core · external vs internal dimensions
The frame area always equals outer area minus inner area, matching l*h - (l-2x)(h-2x).
For an open-top box (or picture frame) of outer dimensions l × h with wall thickness x, the inner cavity is (l − 2x) × (h − 2x). The wall area = outer area − inner area = lh − (l−2x)(h−2x).
What this lesson covers
Try to break it
Drag L for length, H for height, and x for wall thickness. The inner rectangle always measures (L − 2x) by (H − 2x), and the wall (frame) area is always outer area minus inner area. Push x past L/2 or H/2 and the inner core collapses to zero — the box becomes solid wall.
How you build it
Construct a box and its inner cavity.
- Draw the OUTER rectangle of the box (length L by height H).
- Draw the INNER rectangle offset inward by the wall thickness x. The strip between the two rectangles is the wall.
The proof, step by step
Prove that the frame area equals the outer area minus the inner area.
- External volume = l × b × h
- Internal dimensions are reduced by 2x on each side: (l-2x), (b-2x), (h-2x)
- Internal volume = (l-2x)(b-2x)(h-2x)
- Volume of material = External volume - Internal volume
Worked example
A closed wooden box has external dimensions 20 cm × 15 cm × 10 cm. If the wood is 2 cm thick throughout, what is the volume of the wood used?
External volume = 20×15×10 = 3000 cm³. Internal dimensions: (20-4)×(15-4)×(10-4) = 16×11×6 = 1056 cm³. Volume of wood = 3000 - 1056 = 1944 cm³.
- 1944 cm³ — correct
- 1800 cm³
- 2000 cm³
- 1680 cm³