Cuboid and Cube

A cuboid is a 3D box with six rectangular faces , like a brick or a shoebox. A cube is the special case where all sides are equal , like a die. Both shapes have the simplest surface area and volume formulas in 3D geometry.

Definitions

A cuboid has six rectangular faces, twelve edges, and eight vertices. Three independent edge lengths describe it: the length $\ell$, the breadth $b$, and the height $h$. Each pair of opposite faces is congruent.

A cube is a cuboid with $\ell = b = h = a$. All six faces are squares of side $a$.

Formulas

Cuboid.

• Total surface area: $\text{TSA} = 2(\ell b + bh + h\ell)$.
• Lateral surface area (walls only, no top or bottom): $\text{LSA} = 2(\ell + b) h$.
• Volume: $V = \ell \times b \times h$.
• Diagonal: $d = \sqrt{\ell^2 + b^2 + h^2}$.

Cube (special case with $\ell = b = h = a$).

• Total surface area: $\text{TSA} = 6 a^2$.
• Lateral surface area: $\text{LSA} = 4 a^2$.
• Volume: $V = a^3$.
• Diagonal: $d = a\sqrt{3}$.

Why the formulas are right

Cuboid surface area. The six faces come in three pairs: two of $\ell \times b$ (top and bottom), two of $b \times h$ (left and right), two of $h \times \ell$ (front and back). Total: $2 \ell b + 2 b h + 2 h \ell = 2(\ell b + b h + h\ell)$.

Cuboid volume. Imagine stacking unit cubes inside the cuboid. The bottom layer has $\ell \times b$ cubes; the height holds $h$ such layers. Total: $\ell b h$.

Cuboid diagonal. Apply Pythagoras twice. First, the diagonal of the base rectangle is $\sqrt{\ell^2 + b^2}$. Then the body diagonal goes from this corner up to the top, forming a right triangle with the body diagonal as hypotenuse and the base diagonal and height as legs: $d = \sqrt{(\sqrt{\ell^2 + b^2})^2 + h^2} = \sqrt{\ell^2 + b^2 + h^2}$.

Worked examples

Example 1. A cuboid has dimensions $5 \times 4 \times 3$. Find its surface area and volume.

$\text{TSA} = 2(5 \cdot 4 + 4 \cdot 3 + 3 \cdot 5) = 2(20 + 12 + 15) = 2 \cdot 47 = 94$ sq units. $V = 5 \cdot 4 \cdot 3 = 60$ cubic units.

Example 2. A cube has side $6$ cm. Find the total surface area and volume.

$\text{TSA} = 6 \cdot 36 = 216$ cm$^2$. $V = 216$ cm$^3$.

Example 3. The dimensions of a swimming pool are $20 \times 10 \times 2$ m. How much water can it hold?

Volume $= 20 \cdot 10 \cdot 2 = 400$ m$^3$ $= 400000$ litres (since $1$ m$^3$ $= 1000$ litres).

Example 4. A cube of side $7$ cm has surface area $\ldots$ and volume $\ldots$.

$\text{TSA} = 6 \cdot 49 = 294$ cm$^2$. $V = 343$ cm$^3$.

Example 5. The lateral surface area of a cuboidal room of length $5$ m and breadth $4$ m is $90$ m$^2$. Find the height.

$\text{LSA} = 2(\ell + b) h = 2 \cdot 9 \cdot h = 18 h = 90 \Rightarrow h = 5$ m.

Try it yourself

1. Cuboid: $\ell = 8, b = 5, h = 3$. Find TSA, LSA, V.
2. Cube of side $10$. Find TSA, LSA, V.
3. A room has dimensions $6 \times 5 \times 3$ m. Cost of painting walls (LSA only) at Rs. $20$ per m$^2$.
4. A cube has volume $125$. Find its side.
5. A cube has surface area $54$ cm$^2$. Find its side.
6. A swimming pool is $30 \times 20 \times 3$ m. How many litres of water can it hold?
7. Find the diagonal of a cuboid with dimensions $4, 5, 12$.
8. A cuboidal box has TSA $94$ and dimensions $5 \times 4 \times h$. Find $h$.
9. Compare the volume of a $1 \times 2 \times 3$ cuboid with a cube of side $\sqrt[3]{6}$.
10. A cubical tank has volume $8000$ litres. Find its side.

Pitfalls / Insight

• TSA vs. LSA. TSA includes all six faces; LSA includes only the four "side" faces (no top or bottom).
• Units. Volume is in cube units (cm$^3$, m$^3$). Surface area is in square units. Don't mix.
• Cube formulas are special cases of cuboid formulas. With $\ell = b = h = a$, everything reduces.

Insight. The cuboid and cube are the simplest 3D shapes , and their formulas are the most-used in real life: rooms, boxes, tanks, buildings. Master these first.