SURFACE AREA, VOLUME AND CAPACITY (Cuboid, Cube and Cylinder)

199. Cuboid Dimensions · Volume, Surface Area, and Diagonal

The diagonal length matches the formula sqrt(l^2 + b^2 + h^2).

V = l×b×hl = 200l = 200h = 150h = 150b = 150b = 150diag = √(l²+b²+h²) = √(40000+22500+22500) = 291.5diag = √(l²+b²+h²) = √(40000+22500+22500) = 291.5lbh
Cuboid formulas (length l, breadth b, height h): Volume = l × b × h, TSA = 2(lb + bh + hl), LSA = 2h(l + b), Space Diagonal = √(l² + b² + h²). The space diagonal extends Pythagoras to three dimensions.

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Selina ICSE: SURFACE AREA, VOLUME AND CAPACITY (Cuboid, Cube and Cylinder)

What this lesson covers

Try to break it

Drag L, B, and H to extreme values — make the cuboid pancake-flat, pencil-thin, or near-cubical. The diagonal readout always equals √(l² + b² + h²). Try to drag the handles to a setting where the readout disagrees with the formula; you can't.

How you build it

Construct a cuboid.

  • 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 by depth b.
  • 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 (behind A).
  • Mark point F — the back bottom-right corner (behind B).
  • Mark point G — the back top-right corner (behind C).
  • Mark point H — the back top-left corner (behind D).
  • Connect A to E — the first depth edge.
  • Connect B to F.
  • Connect C to G.
  • Connect D to H to complete the cuboid.
  • Draw the space diagonal AG — from front-bottom-left A to back-top-right G. Its length is √(l²+b²+h²).

The proof, step by step

Prove that the diagonal of a cuboid is √(l² + b² + h²).

  • Volume of a cuboid is the space it occupies. It is calculated as length × breadth × height = l × b × h.
  • Total Surface Area (TSA) is the sum of the areas of all 6 faces. TSA = 2(lb + bh + hl).
  • Lateral Surface Area (LSA) is the area of the 4 vertical faces. LSA = 2(l + b)h.
  • The diagonal of a cuboid connects opposite corners. Its length is sqrt(l^2 + b^2 + h^2).

Worked example

A cuboid has length 12 cm, breadth 8 cm, and height 5 cm. What is its volume?

Volume = l × b × h = 12 × 8 × 5 = 480 cm³.

  • 480 cm³ — correct
  • 240 cm³
  • 960 cm³
  • 120 cm³
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